Formula di sommazione di Poisson: differenze tra le versioni

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La funzione è definita sull'asse reale o nello spazio euclideo a <math>n</math> dimensioni. La formula è stata scoperta da [[Siméon Denis Poisson]].</div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>La '''formula di sommazione di Poisson''', anche detta '''risommazione di Poisson''', è un'identità tra due somme infinite, di cui la prima è costruita con una funzione <math>f</math> e la seconda con la sua [[trasformata di Fourier]] <math>\hat f</math>. La funzione è definita sull'asse reale o nello spazio euclideo a <math>n</math> dimensioni. La formula è stata scoperta da [[Siméon Denis Poisson]].</div></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><br /></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><br /></td> </tr> </table><hr class='diff-hr' id='mw-oldid' /> <h2 class='diff-currentversion-title'>Versione delle 08:09, 2 apr 2023</h2> <div class="mw-content-ltr mw-parser-output" lang="it" dir="ltr"><style data-mw-deduplicate="TemplateStyles:r133964453">.mw-parser-output .avviso .mbox-text-div>div,.mw-parser-output .avviso .mbox-text-full-div>div{font-size:90%}.mw-parser-output .avviso .mbox-image{flex-basis:52px;flex-grow:0;flex-shrink:0}.mw-parser-output .avviso .mbox-text-full-div .hide-when-compact{display:block}</style><div style="" class="ambox metadata noprint plainlinks avviso avviso-contenuto"> <div class="avviso-immagine mbox-image noprint"><span typeof="mw:File"><a href="/wiki/File:Question_book_magnify.svg" class="mw-file-description" title="Nessuna nota a piè di pagina"><img alt="Nessuna nota a piè di pagina" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Question_book_magnify.svg/45px-Question_book_magnify.svg.png" decoding="async" width="45" height="45" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Question_book_magnify.svg/68px-Question_book_magnify.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Question_book_magnify.svg/90px-Question_book_magnify.svg.png 2x" data-file-width="60" data-file-height="60" /></a></span></div> <div class="avviso-testo mbox-text"> <div class="mbox-text-div"><b>Questa voce o sezione  sull'argomento matematica è priva o carente di <a href="/wiki/Wikipedia:Uso_delle_fonti" title="Wikipedia:Uso delle fonti">note</a> e <a href="/wiki/Aiuto:Uso_delle_fonti#Citazioni_nel_testo_.28citazioni_interne.29_e_alla_fine" title="Aiuto:Uso delle fonti">riferimenti bibliografici puntuali</a></b>. <div class="hide-when-compact"> <div class="noprint"><hr />Sebbene vi siano una <a href="/wiki/Aiuto:Bibliografia" title="Aiuto:Bibliografia">bibliografia</a> e/o dei <a href="/wiki/Wikipedia:Collegamenti_esterni" title="Wikipedia:Collegamenti esterni">collegamenti esterni</a>, manca la contestualizzazione delle fonti con <a href="/wiki/Aiuto:Note" title="Aiuto:Note">note a piè di pagina</a> o altri riferimenti precisi che indichino puntualmente la provenienza delle informazioni. Puoi <a class="external text" href="https://it.wikipedia.org/w/index.php?title=Formula_di_sommazione_di_Poisson&action=edit">migliorare questa voce</a> <a href="/wiki/Wikipedia:Uso_delle_fonti" title="Wikipedia:Uso delle fonti">citando le fonti</a> più precisamente. Segui i suggerimenti del <a href="/wiki/Progetto:Matematica" title="Progetto:Matematica">progetto di riferimento</a>.</div> </div> </div> </div> </div> <p>La <b>formula di sommazione di Poisson</b>, anche detta <b>risommazione di Poisson</b>, è un'identità tra due somme infinite, di cui la prima è costruita con una funzione <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> e la seconda con la sua <a href="/wiki/Trasformata_di_Fourier" title="Trasformata di Fourier">trasformata di Fourier</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {f}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {f}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14ce989fd75da938ec6f95a0cdb71037b23a11cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.699ex; height:3.176ex;" alt="{\displaystyle {\hat {f}}}"></span>. La funzione è definita sull'asse reale o nello spazio euclideo a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> dimensioni. La formula è stata scoperta da <a href="/wiki/Sim%C3%A9on_Denis_Poisson" class="mw-redirect" title="Siméon Denis Poisson">Siméon Denis Poisson</a>. </p><p>La formula e le sue generalizzazioni sono importanti in molte aree della matematica, tra cui la <a href="/wiki/Teoria_dei_numeri" title="Teoria dei numeri">teoria dei numeri</a>, l'<a href="/wiki/Analisi_armonica" title="Analisi armonica">analisi armonica</a>, e la <a href="/wiki/Geometria_riemanniana" title="Geometria riemanniana">geometria riemanniana</a>. Un modo di interpretare la formula unidimensionale si ottiene osservando la relazione tra lo <a href="/wiki/Spettro_(matematica)" title="Spettro (matematica)">spettro</a> dell'<a href="/wiki/Operatore_di_Laplace-Beltrami" title="Operatore di Laplace-Beltrami">operatore di Laplace-Beltrami</a> sul <a href="/wiki/Cerchio" title="Cerchio">cerchio</a> e la lunghezza delle <a href="/wiki/Geodetica" title="Geodetica">geodetiche</a> periodiche su questa curva. In <a href="/wiki/Analisi_funzionale" title="Analisi funzionale">analisi funzionale</a>, la <a href="/w/index.php?title=Formula_di_Selberg&action=edit&redlink=1" class="new" title="Formula di Selberg (la pagina non esiste)">formula della traccia di Selberg</a> instaura un rapporto di questo tipo - ma di carattere molto più profondo - tra lo spettro del <a href="/wiki/Operatore_di_Laplace" title="Operatore di Laplace">laplaciano</a> e la lunghezza della geodetiche sulle <a href="/wiki/Superficie" title="Superficie">superfici</a> con <a href="/wiki/Curvatura" title="Curvatura">curvatura</a> costante negativa. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="La_formula">La formula</h2></div> <p>Data un'opportuna funzione <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span>, la formula di sommazione di Poisson è data da: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=-\infty }^{\infty }f(n)=\sum _{k=-\infty }^{\infty }{\hat {f}}\left(k\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=-\infty }^{\infty }f(n)=\sum _{k=-\infty }^{\infty }{\hat {f}}\left(k\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ea7b1fc9cff1ab049c148b22d068b1b122933d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:23.705ex; height:7.009ex;" alt="{\displaystyle \sum _{n=-\infty }^{\infty }f(n)=\sum _{k=-\infty }^{\infty }{\hat {f}}\left(k\right)}"></span></dd></dl> <p>dove <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {f}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {f}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14ce989fd75da938ec6f95a0cdb71037b23a11cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.699ex; height:3.176ex;" alt="{\displaystyle {\hat {f}}}"></span> è la <a href="/wiki/Trasformata_di_Fourier" title="Trasformata di Fourier">trasformata di Fourier</a> di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span>, ovvero: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {f}}(\nu )={\mathcal {F}}\{f(x)\}\,{\stackrel {\mathrm {def} }{=}}\int _{-\infty }^{\infty }f(x)\ e^{-2\pi i\nu x}\,dx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ν</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo fence="false" stretchy="false">{</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo>=</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">f</mi> </mrow> </mrow> </mover> </mrow> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> <mi>π</mi> <mi>i</mi> <mi>ν</mi> <mi>x</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {f}}(\nu )={\mathcal {F}}\{f(x)\}\,{\stackrel {\mathrm {def} }{=}}\int _{-\infty }^{\infty }f(x)\ e^{-2\pi i\nu x}\,dx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adae234060d3b53671fdcad9d204db208099d9dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:39.039ex; height:6.009ex;" alt="{\displaystyle {\hat {f}}(\nu )={\mathcal {F}}\{f(x)\}\,{\stackrel {\mathrm {def} }{=}}\int _{-\infty }^{\infty }f(x)\ e^{-2\pi i\nu x}\,dx}"></span></dd></dl> <p>Sostituendo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(xP)\ {\stackrel {\text{def}}{=}}\ f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mi>P</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo>=</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>def</mtext> </mrow> </mover> </mrow> </mrow> <mtext> </mtext> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(xP)\ {\stackrel {\text{def}}{=}}\ f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35db7b978094c068ef1e50f31f01cfce733d41cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.727ex; height:3.843ex;" alt="{\displaystyle g(xP)\ {\stackrel {\text{def}}{=}}\ f(x)}"></span> e sfruttando la proprietà: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}\{g(xP)\}\ ={\frac {1}{P}}\cdot {\hat {g}}\left({\frac {\nu }{P}}\right)\qquad P>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo fence="false" stretchy="false">{</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mi>P</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> <mtext> </mtext> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>P</mi> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>g</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ν</mi> <mi>P</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mspace width="2em" /> <mi>P</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}\{g(xP)\}\ ={\frac {1}{P}}\cdot {\hat {g}}\left({\frac {\nu }{P}}\right)\qquad P>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/796737bcebc1cfffc44f2c822b2491db2b77430e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:36.206ex; height:5.176ex;" alt="{\displaystyle {\mathcal {F}}\{g(xP)\}\ ={\frac {1}{P}}\cdot {\hat {g}}\left({\frac {\nu }{P}}\right)\qquad P>0}"></span></dd></dl> <p>la formula di sommazione diventa: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=-\infty }^{\infty }g(nP)={\frac {1}{P}}\sum _{\nu =-\infty }^{\infty }{\hat {g}}\left({\frac {\nu }{P}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>g</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mi>P</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>P</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>g</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ν</mi> <mi>P</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=-\infty }^{\infty }g(nP)={\frac {1}{P}}\sum _{\nu =-\infty }^{\infty }{\hat {g}}\left({\frac {\nu }{P}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6fd4d09f626a4974894d78112d77fbe6a5ee06e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:30.14ex; height:6.843ex;" alt="{\displaystyle \sum _{n=-\infty }^{\infty }g(nP)={\frac {1}{P}}\sum _{\nu =-\infty }^{\infty }{\hat {g}}\left({\frac {\nu }{P}}\right)}"></span></dd></dl> <p>Definendo inoltre <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s(t+x)\ {\stackrel {\text{def}}{=}}\ g(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>+</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo>=</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>def</mtext> </mrow> </mover> </mrow> </mrow> <mtext> </mtext> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s(t+x)\ {\stackrel {\text{def}}{=}}\ g(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1efa2f8285cce5c2d64316fe5b944b66cabc49d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.473ex; height:3.843ex;" alt="{\displaystyle s(t+x)\ {\stackrel {\text{def}}{=}}\ g(x)}"></span> e utilizzando la proprietà: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}\{s(t+x)\}\ ={\hat {s}}(\nu )\cdot e^{i2\pi \nu t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo fence="false" stretchy="false">{</mo> <mi>s</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>+</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> <mtext> </mtext> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>s</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ν</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mi>ν</mi> <mi>t</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}\{s(t+x)\}\ ={\hat {s}}(\nu )\cdot e^{i2\pi \nu t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34fe5fc6b0f098342041980a469e2ec40e94452" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.964ex; height:3.176ex;" alt="{\displaystyle {\mathcal {F}}\{s(t+x)\}\ ={\hat {s}}(\nu )\cdot e^{i2\pi \nu t}}"></span></dd></dl> <p>si ottiene una rappresentazione periodica di periodo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span>, la cui <a href="/wiki/Serie_di_Fourier" title="Serie di Fourier">serie di Fourier</a> è: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \underbrace {\sum _{n=-\infty }^{\infty }s(t+nP)} _{S_{P}(t)}=\sum _{k=-\infty }^{\infty }\underbrace {{\frac {1}{P}}\cdot {\hat {s}}\left({\frac {k}{P}}\right)} _{S[k]}\ e^{i2\pi {\frac {k}{P}}t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>s</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>+</mo> <mi>n</mi> <mi>P</mi> <mo stretchy="false">)</mo> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </munder> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>P</mi> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>s</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>k</mi> <mi>P</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> <mo stretchy="false">[</mo> <mi>k</mi> <mo stretchy="false">]</mo> </mrow> </munder> <mtext> </mtext> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>k</mi> <mi>P</mi> </mfrac> </mrow> <mi>t</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \underbrace {\sum _{n=-\infty }^{\infty }s(t+nP)} _{S_{P}(t)}=\sum _{k=-\infty }^{\infty }\underbrace {{\frac {1}{P}}\cdot {\hat {s}}\left({\frac {k}{P}}\right)} _{S[k]}\ e^{i2\pi {\frac {k}{P}}t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84bc3afd94cc75fe2e0dd2a617b10d072b78b902" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.005ex; width:42.437ex; height:10.843ex;" alt="{\displaystyle \underbrace {\sum _{n=-\infty }^{\infty }s(t+nP)} _{S_{P}(t)}=\sum _{k=-\infty }^{\infty }\underbrace {{\frac {1}{P}}\cdot {\hat {s}}\left({\frac {k}{P}}\right)} _{S[k]}\ e^{i2\pi {\frac {k}{P}}t}}"></span></dd></dl> <p>Si può mostrare che tale relazione vale nel senso che se <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s(t)\in L^{1}(\mathbb {R} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s(t)\in L^{1}(\mathbb {R} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f5c7003bbb16b5a6e043b14278841923b963f38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.704ex; height:3.176ex;" alt="{\displaystyle s(t)\in L^{1}(\mathbb {R} )}"></span> allora il membro alla destra è la serie di Fourier del membro alla sinistra, e tale serie può divergere. Infatti, dal <a href="/wiki/Teorema_della_convergenza_dominata" title="Teorema della convergenza dominata">teorema della convergenza dominata</a> segue che la somma <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{P}(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{P}(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf71861ff7b3baf5f18412dbe7aff81890d36633" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.206ex; height:2.843ex;" alt="{\displaystyle s_{P}(t)}"></span> esiste ed è finita per quasi tutti i valori di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"></span>, ed è integrabile sull'intervallo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,P]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>P</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,P]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e22a95e69fea5905acab328644408c110eedea0e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.236ex; height:2.843ex;" alt="{\displaystyle [0,P]}"></span>. Inoltre, dall'espressione del membro alla destra si evince che è sufficiente mostrare che i coefficienti di tale serie di Fourier sono <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle {\frac {1}{P}}{\hat {s}}\left({\frac {k}{P}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>P</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>s</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>k</mi> <mi>P</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\frac {1}{P}}{\hat {s}}\left({\frac {k}{P}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3acf0629d810edb8ec52e68d9c67bd054d7b999" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:6.59ex; height:3.509ex;" alt="{\displaystyle \scriptstyle {\frac {1}{P}}{\hat {s}}\left({\frac {k}{P}}\right)}"></span>, procedendo come segue: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}S[k]\ &{\stackrel {\text{def}}{=}}\ {\frac {1}{P}}\int _{0}^{P}s_{P}(t)\cdot e^{-i2\pi {\frac {k}{P}}t}\,dt\\&=\ {\frac {1}{P}}\int _{0}^{P}\left(\sum _{n=-\infty }^{\infty }s(t+nP)\right)\cdot e^{-i2\pi {\frac {k}{P}}t}\,dt\\&=\ {\frac {1}{P}}\sum _{n=-\infty }^{\infty }\int _{0}^{P}s(t+nP)\cdot e^{-i2\pi {\frac {k}{P}}t}\,dt\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>S</mi> <mo stretchy="false">[</mo> <mi>k</mi> <mo stretchy="false">]</mo> <mtext> </mtext> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo>=</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>def</mtext> </mrow> </mover> </mrow> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>P</mi> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msubsup> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>k</mi> <mi>P</mi> </mfrac> </mrow> <mi>t</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>P</mi> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msubsup> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>s</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>+</mo> <mi>n</mi> <mi>P</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mo>⋅</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>k</mi> <mi>P</mi> </mfrac> </mrow> <mi>t</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>P</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msubsup> <mi>s</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>+</mo> <mi>n</mi> <mi>P</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>k</mi> <mi>P</mi> </mfrac> </mrow> <mi>t</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}S[k]\ &{\stackrel {\text{def}}{=}}\ {\frac {1}{P}}\int _{0}^{P}s_{P}(t)\cdot e^{-i2\pi {\frac {k}{P}}t}\,dt\\&=\ {\frac {1}{P}}\int _{0}^{P}\left(\sum _{n=-\infty }^{\infty }s(t+nP)\right)\cdot e^{-i2\pi {\frac {k}{P}}t}\,dt\\&=\ {\frac {1}{P}}\sum _{n=-\infty }^{\infty }\int _{0}^{P}s(t+nP)\cdot e^{-i2\pi {\frac {k}{P}}t}\,dt\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c35550db65dab80998d1f028cdaffbc8430ebab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.722ex; margin-bottom: -0.282ex; width:46.756ex; height:21.176ex;" alt="{\displaystyle {\begin{aligned}S[k]\ &{\stackrel {\text{def}}{=}}\ {\frac {1}{P}}\int _{0}^{P}s_{P}(t)\cdot e^{-i2\pi {\frac {k}{P}}t}\,dt\\&=\ {\frac {1}{P}}\int _{0}^{P}\left(\sum _{n=-\infty }^{\infty }s(t+nP)\right)\cdot e^{-i2\pi {\frac {k}{P}}t}\,dt\\&=\ {\frac {1}{P}}\sum _{n=-\infty }^{\infty }\int _{0}^{P}s(t+nP)\cdot e^{-i2\pi {\frac {k}{P}}t}\,dt\end{aligned}}}"></span></dd></dl> <p>dove lo scambio tra la somma e l'integrale è ancora permesso dal teorema della convergenza dominata. Con un'integrazione per sostituzione, ponendo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau =t+nPt}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> <mo>=</mo> <mi>t</mi> <mo>+</mo> <mi>n</mi> <mi>P</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau =t+nPt}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cb2b4c1c9fea98443a9d5d644d4cc4defec56e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.96ex; height:2.343ex;" alt="{\displaystyle \tau =t+nPt}"></span>, la precedente espressione diventa infine: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}S[k]={\frac {1}{P}}\sum _{n=-\infty }^{\infty }\int _{nP}^{nP+P}s(\tau )\ e^{-i2\pi {\frac {k}{P}}\tau }\ \underbrace {e^{i2\pi kn}} _{1}\,d\tau \ =\ {\frac {1}{P}}\int _{-\infty }^{\infty }s(\tau )\ e^{-i2\pi {\frac {k}{P}}\tau }d\tau ={\frac {1}{P}}\cdot {\hat {s}}\left({\frac {k}{P}}\right)\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>S</mi> <mo stretchy="false">[</mo> <mi>k</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>P</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mi>P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mi>P</mi> <mo>+</mo> <mi>P</mi> </mrow> </msubsup> <mi>s</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>k</mi> <mi>P</mi> </mfrac> </mrow> <mi>τ</mi> </mrow> </msup> <mtext> </mtext> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mi>k</mi> <mi>n</mi> </mrow> </msup> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </munder> <mspace width="thinmathspace" /> <mi>d</mi> <mi>τ</mi> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>P</mi> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>s</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>k</mi> <mi>P</mi> </mfrac> </mrow> <mi>τ</mi> </mrow> </msup> <mi>d</mi> <mi>τ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>P</mi> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>s</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>k</mi> <mi>P</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}S[k]={\frac {1}{P}}\sum _{n=-\infty }^{\infty }\int _{nP}^{nP+P}s(\tau )\ e^{-i2\pi {\frac {k}{P}}\tau }\ \underbrace {e^{i2\pi kn}} _{1}\,d\tau \ =\ {\frac {1}{P}}\int _{-\infty }^{\infty }s(\tau )\ e^{-i2\pi {\frac {k}{P}}\tau }d\tau ={\frac {1}{P}}\cdot {\hat {s}}\left({\frac {k}{P}}\right)\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f28c18bbdaa3ce84b12482eb46e64c6b7e54417c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:87.2ex; height:7.509ex;" alt="{\displaystyle {\begin{aligned}S[k]={\frac {1}{P}}\sum _{n=-\infty }^{\infty }\int _{nP}^{nP+P}s(\tau )\ e^{-i2\pi {\frac {k}{P}}\tau }\ \underbrace {e^{i2\pi kn}} _{1}\,d\tau \ =\ {\frac {1}{P}}\int _{-\infty }^{\infty }s(\tau )\ e^{-i2\pi {\frac {k}{P}}\tau }d\tau ={\frac {1}{P}}\cdot {\hat {s}}\left({\frac {k}{P}}\right)\end{aligned}}}"></span></dd></dl> <p>In modo analogo, la rappresentazione periodica della trasformata di Fourier di una funzione possiede un equivalente sviluppo in serie di Fourier: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{k=-\infty }^{\infty }{\hat {s}}(\nu +k/T)=\sum _{n=-\infty }^{\infty }T\cdot s(nT)\ e^{-i2\pi nT\nu }\equiv {\mathcal {F}}\left\{\sum _{n=-\infty }^{\infty }T\cdot s(nT)\ \delta (t-nT)\right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>s</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ν</mi> <mo>+</mo> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>T</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>T</mi> <mo>⋅</mo> <mi>s</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mi>T</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mi>n</mi> <mi>T</mi> <mi>ν</mi> </mrow> </msup> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow> <mo>{</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>T</mi> <mo>⋅</mo> <mi>s</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mi>T</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>n</mi> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k=-\infty }^{\infty }{\hat {s}}(\nu +k/T)=\sum _{n=-\infty }^{\infty }T\cdot s(nT)\ e^{-i2\pi nT\nu }\equiv {\mathcal {F}}\left\{\sum _{n=-\infty }^{\infty }T\cdot s(nT)\ \delta (t-nT)\right\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f95e22ec8f82090e169f59d913660bdfcfadfaf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:77.194ex; height:7.509ex;" alt="{\displaystyle \sum _{k=-\infty }^{\infty }{\hat {s}}(\nu +k/T)=\sum _{n=-\infty }^{\infty }T\cdot s(nT)\ e^{-i2\pi nT\nu }\equiv {\mathcal {F}}\left\{\sum _{n=-\infty }^{\infty }T\cdot s(nT)\ \delta (t-nT)\right\}}"></span></dd></dl> <p>dove <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> è l'intervallo temporale che corrisponde al periodo al quale <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c484de351ba40ccb9a5ad522c29c1aac5686c0df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.739ex; height:2.843ex;" alt="{\displaystyle s(t)}"></span> viene campionata. </p> <div class="mw-heading mw-heading2"><h2 id="Teorema">Teorema</h2></div> <p>Sia <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> una funzione complessa definita su <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span> due volte continuamente <a href="/wiki/Funzione_differenziabile" title="Funzione differenziabile">differenziabile</a>, le cui prime due derivate su <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span> siano integrabili, e che soddisfi la relazione: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |f(x)|\leq {\frac {C}{1+x^{2}}}\qquad \forall x\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>C</mi> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mspace width="2em" /> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |f(x)|\leq {\frac {C}{1+x^{2}}}\qquad \forall x\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d214300226179fa79a573c3939c41a06e1cd643" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:27.819ex; height:5.843ex;" alt="{\displaystyle |f(x)|\leq {\frac {C}{1+x^{2}}}\qquad \forall x\in \mathbb {R} }"></span></dd></dl> <p>Sia inoltre <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> un numero strettamente positivo. Detto <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{0}=2\pi /a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{0}=2\pi /a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c420e774f94076694b51c80d7553a40cea8e848f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.485ex; height:2.843ex;" alt="{\displaystyle \omega _{0}=2\pi /a}"></span> il modo fondamentale, vale la seguente identità: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(t)\equiv \sum _{n=-\infty }^{\infty }f(t+na)={\frac {1}{a}}\sum _{m=-\infty }^{\infty }{\hat {f}}(m\omega _{0})\ e^{im\omega _{0}t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>≡</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>+</mo> <mi>n</mi> <mi>a</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>a</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>m</mi> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mtext> </mtext> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>m</mi> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>t</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S(t)\equiv \sum _{n=-\infty }^{\infty }f(t+na)={\frac {1}{a}}\sum _{m=-\infty }^{\infty }{\hat {f}}(m\omega _{0})\ e^{im\omega _{0}t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2493503b064b3ebe4c80c99afd39da4d3f7c8851" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:48.197ex; height:6.843ex;" alt="{\displaystyle S(t)\equiv \sum _{n=-\infty }^{\infty }f(t+na)={\frac {1}{a}}\sum _{m=-\infty }^{\infty }{\hat {f}}(m\omega _{0})\ e^{im\omega _{0}t}}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Dimostrazione">Dimostrazione</h3></div> <p>Il lato sinistro della formula sommatoria di Poisson è la somma di una serie di funzioni continue. L'ipotesi fatte circa il comportamento di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> all'infinito implica che la serie <a href="/w/index.php?title=Convergenza_in_norma&action=edit&redlink=1" class="new" title="Convergenza in norma (la pagina non esiste)">converge normalmente</a> su ogni compatto <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [-a,a]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mo>−</mo> <mi>a</mi> <mo>,</mo> <mi>a</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [-a,a]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50ccbcece37f9ec0a4c6d396be3a143a0b76d5c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.595ex; height:2.843ex;" alt="{\displaystyle [-a,a]}"></span> di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span>. Pertanto, la sua somma è una funzione continua, e la formula di definizione mostra che è periodica di periodo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span>. </p><p>Quindi si possono calcolare i coefficienti della sua <a href="/wiki/Serie_di_Fourier" title="Serie di Fourier">serie di Fourier</a> a esponenziali sul <a href="/wiki/Base_ortonormale" title="Base ortonormale">sistema ortonormale</a> <a href="/wiki/Spazio_completo" class="mw-redirect" title="Spazio completo">completo</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\{e^{i2\pi {\frac {n}{a}}}\right\}_{n=-\infty }^{n=+\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow> <mo>{</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mi>a</mi> </mfrac> </mrow> </mrow> </msup> <mo>}</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{e^{i2\pi {\frac {n}{a}}}\right\}_{n=-\infty }^{n=+\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aee781b46d35b31fd75a35572cda0359416001a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:13.803ex; height:5.176ex;" alt="{\displaystyle \left\{e^{i2\pi {\frac {n}{a}}}\right\}_{n=-\infty }^{n=+\infty }}"></span>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{m}=\int _{0}^{a}\sum _{n\in \mathbb {Z} }f(t+na)e^{-2\mathrm {i} \pi mt/a}\,dt}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msubsup> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mrow> </munder> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>+</mo> <mi>n</mi> <mi>a</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mi>π</mi> <mi>m</mi> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>a</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{m}=\int _{0}^{a}\sum _{n\in \mathbb {Z} }f(t+na)e^{-2\mathrm {i} \pi mt/a}\,dt}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c30c1d311e634eae50615d78900727ad698c7f06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:33.854ex; height:6.676ex;" alt="{\displaystyle c_{m}=\int _{0}^{a}\sum _{n\in \mathbb {Z} }f(t+na)e^{-2\mathrm {i} \pi mt/a}\,dt}"></span></dd></dl> <p>Grazie alla convergenza normale della serie definente <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span> possiamo scambiare somma e integrazione, e scrivere quindi: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{m}=\sum _{n\in \mathbb {Z} }\int _{0}^{a}f(t+na)e^{-2\mathrm {i} \pi mt/a}\,dt}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mrow> </munder> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>+</mo> <mi>n</mi> <mi>a</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mi>π</mi> <mi>m</mi> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>a</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{m}=\sum _{n\in \mathbb {Z} }\int _{0}^{a}f(t+na)e^{-2\mathrm {i} \pi mt/a}\,dt}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ad9947200427f029673c474a92020130640a44b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:33.854ex; height:6.676ex;" alt="{\displaystyle c_{m}=\sum _{n\in \mathbb {Z} }\int _{0}^{a}f(t+na)e^{-2\mathrm {i} \pi mt/a}\,dt}"></span></dd></dl> <p>Se si effettua in ogni integrale il cambio di variabile <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s=t+na}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>=</mo> <mi>t</mi> <mo>+</mo> <mi>n</mi> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s=t+na}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fdd1d845b31a34a938534c1406769846325dd30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.493ex; height:2.176ex;" alt="{\displaystyle s=t+na}"></span> si ottiene: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{m}=\sum _{n\in \mathbb {Z} }\int _{na}^{(n+1)a}f(s)e^{-2\mathrm {i} \pi m(s-na)/a}\,ds={\hat {f}}(2m\pi /a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mrow> </munder> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>a</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mi>π</mi> <mi>m</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo>−</mo> <mi>n</mi> <mi>a</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>a</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>s</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>m</mi> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{m}=\sum _{n\in \mathbb {Z} }\int _{na}^{(n+1)a}f(s)e^{-2\mathrm {i} \pi m(s-na)/a}\,ds={\hat {f}}(2m\pi /a)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0fc964702ee0fac57086e51f65b82af08ef4989" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:51.382ex; height:7.176ex;" alt="{\displaystyle c_{m}=\sum _{n\in \mathbb {Z} }\int _{na}^{(n+1)a}f(s)e^{-2\mathrm {i} \pi m(s-na)/a}\,ds={\hat {f}}(2m\pi /a)}"></span></dd></dl> <p>Dalle ipotesi su <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> e le sue derivate, e dalle identità classiche sulle trasformata della derivata, si vede che la funzione <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {f}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {f}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14ce989fd75da938ec6f95a0cdb71037b23a11cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.699ex; height:3.176ex;" alt="{\displaystyle {\hat {f}}}"></span> soddisfa la relazione: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall \omega \in \mathbb {R} ,\quad |{\hat {f}}(\omega )|\leq {\hat {C}}/(1+\omega ^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>ω</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>,</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>C</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall \omega \in \mathbb {R} ,\quad |{\hat {f}}(\omega )|\leq {\hat {C}}/(1+\omega ^{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc9dc84736a0cb95815c252da31ec794ccb711f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.201ex; height:3.343ex;" alt="{\displaystyle \forall \omega \in \mathbb {R} ,\quad |{\hat {f}}(\omega )|\leq {\hat {C}}/(1+\omega ^{2})}"></span></dd></dl> <p>Pertanto, la serie di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5a92f980a7ccf6827b6925c6d6421984d9c5859" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.682ex; height:2.009ex;" alt="{\displaystyle c_{m}}"></span> è assolutamente convergente ci troviamo in una situazione in cui si può sommare la serie di Fourier di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span>, e ottenere: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(t)={\frac {1}{a}}\sum _{m\in \mathbb {Z} }c_{m}e^{2\mathrm {i} \pi mt/a}={\frac {1}{a}}\sum _{m\in \mathbb {Z} }{\hat {f}}(2m\pi /a)e^{2\mathrm {i} \pi mt/a}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>a</mi> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mrow> </munder> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mi>π</mi> <mi>m</mi> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>a</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>a</mi> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>m</mi> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>a</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mi>π</mi> <mi>m</mi> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>a</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S(t)={\frac {1}{a}}\sum _{m\in \mathbb {Z} }c_{m}e^{2\mathrm {i} \pi mt/a}={\frac {1}{a}}\sum _{m\in \mathbb {Z} }{\hat {f}}(2m\pi /a)e^{2\mathrm {i} \pi mt/a}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dead6cef0898e2a313d88c2574e59ccd664f2b3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:50.943ex; height:6.509ex;" alt="{\displaystyle S(t)={\frac {1}{a}}\sum _{m\in \mathbb {Z} }c_{m}e^{2\mathrm {i} \pi mt/a}={\frac {1}{a}}\sum _{m\in \mathbb {Z} }{\hat {f}}(2m\pi /a)e^{2\mathrm {i} \pi mt/a}}"></span></dd></dl> <p>Questa è la formula desiderata, ricordando che <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi /a=\omega _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>a</mi> <mo>=</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi /a=\omega _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8bc8b83362469117563bd6c2cefa2372b763b29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.485ex; height:2.843ex;" alt="{\displaystyle 2\pi /a=\omega _{0}}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Teoria_delle_distribuzioni">Teoria delle distribuzioni</h2></div> <style data-mw-deduplicate="TemplateStyles:r130657691">body:not(.skin-minerva) .mw-parser-output .vedi-anche{font-size:95%}</style><style data-mw-deduplicate="TemplateStyles:r139142988">.mw-parser-output .hatnote-content{align-items:center;display:flex}.mw-parser-output .hatnote-icon{flex-shrink:0}.mw-parser-output .hatnote-icon img{display:flex}.mw-parser-output .hatnote-text{font-style:italic}body:not(.skin-minerva) .mw-parser-output .hatnote{border:1px solid #CCC;display:flex;margin:.5em 0;padding:.2em .5em}body:not(.skin-minerva) .mw-parser-output .hatnote-text{padding-left:.5em}body.skin-minerva .mw-parser-output .hatnote-icon{padding-right:8px}body.skin-minerva .mw-parser-output .hatnote-icon img{height:auto;width:16px}body.skin--responsive .mw-parser-output .hatnote a.new{color:#d73333}body.skin--responsive .mw-parser-output .hatnote a.new:visited{color:#a55858}</style> <div class="hatnote noprint vedi-anche"> <div class="hatnote-content"><span class="noviewer hatnote-icon" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/18px-Magnifying_glass_icon_mgx2.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/27px-Magnifying_glass_icon_mgx2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/36px-Magnifying_glass_icon_mgx2.svg.png 2x" data-file-width="286" data-file-height="280" /></span></span> <span class="hatnote-text">Lo stesso argomento in dettaglio: <b><a href="/wiki/Distribuzione_(matematica)" title="Distribuzione (matematica)">Distribuzione (matematica)</a></b>.</span></div> </div> <p>Un modo comodo per aggirare le condizioni di regolarità imposte alla funzione <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> è di collocare la formula nel contesto più ampio della teoria delle distribuzioni. Se <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta (x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta (x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4457507451c205a7e6adda92d919ee4c4a369cea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.188ex; height:2.843ex;" alt="{\displaystyle \delta (x)}"></span> è la <a href="/wiki/Delta_di_Dirac" title="Delta di Dirac">distribuzione di Dirac</a> e se si introduce la seguente distribuzione, nota come <a href="/wiki/Pettine_di_Dirac" title="Pettine di Dirac">pettine di Dirac</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta (x)\equiv \sum _{n\in \mathbb {Z} }\delta (x-n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≡</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mrow> </munder> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta (x)\equiv \sum _{n\in \mathbb {Z} }\delta (x-n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e8e24f40f24a2bf7d2674323d3e45c32c71ba11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:20.338ex; height:5.676ex;" alt="{\displaystyle \Delta (x)\equiv \sum _{n\in \mathbb {Z} }\delta (x-n)}"></span></dd></dl> <p>Un modo elegante per riscrivere la somma equivale a dire che <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta (x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta (x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1dbdf957b843d924706374d3094a44847a072c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.075ex; height:2.843ex;" alt="{\displaystyle \Delta (x)}"></span> è la trasformata di Fourier di sé stessa. </p><p>Si consideri una distribuzione <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> le cui derivate siano a decrescenza rapida. Considerando il pettine di Dirac e il suo sviluppo in <a href="/wiki/Serie_di_Fourier" title="Serie di Fourier">serie di Fourier</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=-\infty }^{\infty }\delta (x-nP)\equiv \sum _{k=-\infty }^{\infty }{\frac {1}{P}}\cdot e^{-i2\pi {\frac {k}{P}}x}\quad {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\quad {\frac {1}{P}}\cdot \sum _{k=-\infty }^{\infty }\delta (\nu +k/P)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>n</mi> <mi>P</mi> <mo stretchy="false">)</mo> <mo>≡</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>P</mi> </mfrac> </mrow> <mo>⋅</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>k</mi> <mi>P</mi> </mfrac> </mrow> <mi>x</mi> </mrow> </msup> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo stretchy="false">⟺</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mover> </mrow> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>P</mi> </mfrac> </mrow> <mo>⋅</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>ν</mi> <mo>+</mo> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>P</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=-\infty }^{\infty }\delta (x-nP)\equiv \sum _{k=-\infty }^{\infty }{\frac {1}{P}}\cdot e^{-i2\pi {\frac {k}{P}}x}\quad {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\quad {\frac {1}{P}}\cdot \sum _{k=-\infty }^{\infty }\delta (\nu +k/P)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ce0cb121c437e9713b67a1316dfeaa7feb1fb36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:65.964ex; height:7.009ex;" alt="{\displaystyle \sum _{n=-\infty }^{\infty }\delta (x-nP)\equiv \sum _{k=-\infty }^{\infty }{\frac {1}{P}}\cdot e^{-i2\pi {\frac {k}{P}}x}\quad {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\quad {\frac {1}{P}}\cdot \sum _{k=-\infty }^{\infty }\delta (\nu +k/P)}"></span></dd></dl> <p>Si ha: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\sum _{k=-\infty }^{\infty }{\hat {f}}(k)&=\sum _{k=-\infty }^{\infty }\left(\int _{-\infty }^{\infty }f(x)\ e^{-i2\pi kx}dx\right)=\int _{-\infty }^{\infty }f(x)\underbrace {\left(\sum _{k=-\infty }^{\infty }e^{-i2\pi kx}\right)} _{\sum _{n=-\infty }^{\infty }\delta (x-n)}dx\\&=\sum _{n=-\infty }^{\infty }\left(\int _{-\infty }^{\infty }f(x)\ \delta (x-n)\ dx\right)=\sum _{n=-\infty }^{\infty }f(n)\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow> <mo>(</mo> <mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mi>k</mi> <mi>x</mi> </mrow> </msup> <mi>d</mi> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mi>k</mi> <mi>x</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </munder> <mi>d</mi> <mi>x</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow> <mo>(</mo> <mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mi>d</mi> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sum _{k=-\infty }^{\infty }{\hat {f}}(k)&=\sum _{k=-\infty }^{\infty }\left(\int _{-\infty }^{\infty }f(x)\ e^{-i2\pi kx}dx\right)=\int _{-\infty }^{\infty }f(x)\underbrace {\left(\sum _{k=-\infty }^{\infty }e^{-i2\pi kx}\right)} _{\sum _{n=-\infty }^{\infty }\delta (x-n)}dx\\&=\sum _{n=-\infty }^{\infty }\left(\int _{-\infty }^{\infty }f(x)\ \delta (x-n)\ dx\right)=\sum _{n=-\infty }^{\infty }f(n)\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6681131c13d95281d693bd19857249ea68a4ba18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.838ex; width:72.72ex; height:18.843ex;" alt="{\displaystyle {\begin{aligned}\sum _{k=-\infty }^{\infty }{\hat {f}}(k)&=\sum _{k=-\infty }^{\infty }\left(\int _{-\infty }^{\infty }f(x)\ e^{-i2\pi kx}dx\right)=\int _{-\infty }^{\infty }f(x)\underbrace {\left(\sum _{k=-\infty }^{\infty }e^{-i2\pi kx}\right)} _{\sum _{n=-\infty }^{\infty }\delta (x-n)}dx\\&=\sum _{n=-\infty }^{\infty }\left(\int _{-\infty }^{\infty }f(x)\ \delta (x-n)\ dx\right)=\sum _{n=-\infty }^{\infty }f(n)\end{aligned}}}"></span></dd></dl> <p>e similmente: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\sum _{k=-\infty }^{\infty }{\hat {s}}(\nu +k/T)&=\sum _{k=-\infty }^{\infty }{\mathcal {F}}\left\{s(t)\cdot e^{-i2\pi {\frac {k}{T}}t}\right\}\\&={\mathcal {F}}{\bigg \{}s(t)\underbrace {\sum _{k=-\infty }^{\infty }e^{-i2\pi {\frac {k}{T}}t}} _{T\sum _{n=-\infty }^{\infty }\delta (t-nT)}{\bigg \}}={\mathcal {F}}\left\{\sum _{n=-\infty }^{\infty }T\cdot s(nT)\cdot \delta (t-nT)\right\}\\&=\sum _{n=-\infty }^{\infty }T\cdot s(nT)\cdot {\mathcal {F}}\left\{\delta (t-nT)\right\}=\sum _{n=-\infty }^{\infty }T\cdot s(nT)\cdot e^{-i2\pi nT\nu }\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>s</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ν</mi> <mo>+</mo> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>T</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow> <mo>{</mo> <mrow> <mi>s</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>k</mi> <mi>T</mi> </mfrac> </mrow> <mi>t</mi> </mrow> </msup> </mrow> <mo>}</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">{</mo> </mrow> </mrow> <mi>s</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>k</mi> <mi>T</mi> </mfrac> </mrow> <mi>t</mi> </mrow> </msup> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>n</mi> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">}</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow> <mo>{</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>T</mi> <mo>⋅</mo> <mi>s</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mi>T</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>n</mi> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> <mo>}</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>T</mi> <mo>⋅</mo> <mi>s</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mi>T</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow> <mo>{</mo> <mrow> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>n</mi> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> <mo>}</mo> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>T</mi> <mo>⋅</mo> <mi>s</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mi>T</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mi>n</mi> <mi>T</mi> <mi>ν</mi> </mrow> </msup> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sum _{k=-\infty }^{\infty }{\hat {s}}(\nu +k/T)&=\sum _{k=-\infty }^{\infty }{\mathcal {F}}\left\{s(t)\cdot e^{-i2\pi {\frac {k}{T}}t}\right\}\\&={\mathcal {F}}{\bigg \{}s(t)\underbrace {\sum _{k=-\infty }^{\infty }e^{-i2\pi {\frac {k}{T}}t}} _{T\sum _{n=-\infty }^{\infty }\delta (t-nT)}{\bigg \}}={\mathcal {F}}\left\{\sum _{n=-\infty }^{\infty }T\cdot s(nT)\cdot \delta (t-nT)\right\}\\&=\sum _{n=-\infty }^{\infty }T\cdot s(nT)\cdot {\mathcal {F}}\left\{\delta (t-nT)\right\}=\sum _{n=-\infty }^{\infty }T\cdot s(nT)\cdot e^{-i2\pi nT\nu }\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5fd38c833be1befd557e7bc403382338eefa7a66" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -12.505ex; width:78.933ex; height:26.176ex;" alt="{\displaystyle {\begin{aligned}\sum _{k=-\infty }^{\infty }{\hat {s}}(\nu +k/T)&=\sum _{k=-\infty }^{\infty }{\mathcal {F}}\left\{s(t)\cdot e^{-i2\pi {\frac {k}{T}}t}\right\}\\&={\mathcal {F}}{\bigg \{}s(t)\underbrace {\sum _{k=-\infty }^{\infty }e^{-i2\pi {\frac {k}{T}}t}} _{T\sum _{n=-\infty }^{\infty }\delta (t-nT)}{\bigg \}}={\mathcal {F}}\left\{\sum _{n=-\infty }^{\infty }T\cdot s(nT)\cdot \delta (t-nT)\right\}\\&=\sum _{n=-\infty }^{\infty }T\cdot s(nT)\cdot {\mathcal {F}}\left\{\delta (t-nT)\right\}=\sum _{n=-\infty }^{\infty }T\cdot s(nT)\cdot e^{-i2\pi nT\nu }\end{aligned}}}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Somma_periodica">Somma periodica</h2></div> <p>Una forma della sommazione di Poisson si ottiene considerando una <a href="/wiki/Funzione_periodica" title="Funzione periodica">funzione periodica</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{P}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{P}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43bdbe4ab8c7bbbb89a5410c25b536d10befbb5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.606ex; height:2.509ex;" alt="{\displaystyle f_{P}}"></span> di periodo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> e rappresendola attraverso un funzione <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> non periodica nel seguente modo: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{P}(x)=\sum _{n=-\infty }^{\infty }f(x+nP)=\sum _{n=-\infty }^{\infty }f(x-nP)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>n</mi> <mi>P</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>n</mi> <mi>P</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{P}(x)=\sum _{n=-\infty }^{\infty }f(x+nP)=\sum _{n=-\infty }^{\infty }f(x-nP)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/704f5385db677c0a4de8f3f6be7ef8542c03bb67" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:43.885ex; height:6.843ex;" alt="{\displaystyle f_{P}(x)=\sum _{n=-\infty }^{\infty }f(x+nP)=\sum _{n=-\infty }^{\infty }f(x-nP)}"></span></dd></dl> <p>Tale espressione è detta <i>sommazione periodica</i>, e se <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{P}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{P}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43bdbe4ab8c7bbbb89a5410c25b536d10befbb5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.606ex; height:2.509ex;" alt="{\displaystyle f_{P}}"></span> è rappresentabile in <a href="/wiki/Serie_di_Fourier" title="Serie di Fourier">serie di Fourier</a> complessa i coefficienti di tale serie sono proporzionali ai valori della <a href="/wiki/Trasformata_di_Fourier" title="Trasformata di Fourier">trasformata di Fourier</a> di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> "campionata" ad intervalli <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle 1/P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>P</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle 1/P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1460b1ac04bd11a17f0b7280f8cb9b9231c9c17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.878ex; height:2.176ex;" alt="{\displaystyle \scriptstyle 1/P}"></span>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>In modo analogo, una serie di Fourier i cui coefficienti sono ottenuti campionando <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> è equivalente alla somma periodica della trasformata di Fourier di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span>, nota come <a href="/wiki/Trasformata_di_Fourier_discreta" class="mw-redirect" title="Trasformata di Fourier discreta">trasformata di Fourier discreta</a>. </p><p>Se si rappresenta una funzione periodica utilizzando il dominio <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} /(P\cdot \mathbb {Z} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>P</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} /(P\cdot \mathbb {Z} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a413f5d22be4cba49649249d97ca90afe14f513" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.625ex; height:2.843ex;" alt="{\displaystyle \mathbb {R} /(P\cdot \mathbb {Z} )}"></span> (<a href="/wiki/Spazio_vettoriale_quoziente" title="Spazio vettoriale quoziente">spazio quoziente</a>) si può scrivere: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{P}:\mathbb {R} /(P\cdot \mathbb {Z} )\to \mathbb {R} \qquad \varphi _{P}(x)=\sum _{\tau \in x}f(\tau )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>P</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mspace width="2em" /> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>τ</mi> <mo>∈</mo> <mi>x</mi> </mrow> </munder> <mi>f</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi _{P}:\mathbb {R} /(P\cdot \mathbb {Z} )\to \mathbb {R} \qquad \varphi _{P}(x)=\sum _{\tau \in x}f(\tau )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e404aae51553a5571d012740d92b3032cf51f66" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:41.742ex; height:5.509ex;" alt="{\displaystyle \varphi _{P}:\mathbb {R} /(P\cdot \mathbb {Z} )\to \mathbb {R} \qquad \varphi _{P}(x)=\sum _{\tau \in x}f(\tau )}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Applicazioni_della_risommazione_di_Poisson">Applicazioni della risommazione di Poisson</h2></div> <p>Un risultato di fondamentale importanza della formula di sommazione è fornire un <a href="/wiki/Teorema_del_campionamento_di_Nyquist-Shannon#Formula_di_sommazione_di_Poisson" title="Teorema del campionamento di Nyquist-Shannon">criterio</a> che garantisca la ricostruibilità di un segnale <a href="/wiki/Campionamento_(teoria_dei_segnali)" title="Campionamento (teoria dei segnali)">campionato</a>. Essa lega i campioni di una generica <a href="/wiki/Forma_d%27onda" title="Forma d'onda">forma d'onda</a> nel <a href="/wiki/Dominio_del_tempo" title="Dominio del tempo">dominio del tempo</a> alle ripetizioni della sua trasformata nel <a href="/wiki/Dominio_della_frequenza" title="Dominio della frequenza">dominio della frequenza</a>: scegliendo un intervallo di campionamento sufficientemente rapido non vi saranno sovrapposizioni nel dominio della frequenza e sarà sempre possibile ricostruire il segnale campionato. </p><p>La sommazione è inoltre utile per determinare la somma di serie come: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\equiv \sum _{n=1}^{\infty }{\frac {1}{n^{2}}}={\frac {\pi ^{2}}{6}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>≡</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>6</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\equiv \sum _{n=1}^{\infty }{\frac {1}{n^{2}}}={\frac {\pi ^{2}}{6}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb3804c8284c51e69a3fa93b7a9dd3d995e28d95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:17.948ex; height:6.843ex;" alt="{\displaystyle S\equiv \sum _{n=1}^{\infty }{\frac {1}{n^{2}}}={\frac {\pi ^{2}}{6}}}"></span></dd></dl> <p>o anche: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\equiv -\sum _{n=1}^{\infty }{\frac {(-1)^{n}}{n^{4}}}={\frac {7\pi ^{4}}{720}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>≡</mo> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>7</mn> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> <mn>720</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\equiv -\sum _{n=1}^{\infty }{\frac {(-1)^{n}}{n^{4}}}={\frac {7\pi ^{4}}{720}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84d700cf3bad27d06b1d3479c5a9264cf584152b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:24.855ex; height:6.843ex;" alt="{\displaystyle S\equiv -\sum _{n=1}^{\infty }{\frac {(-1)^{n}}{n^{4}}}={\frac {7\pi ^{4}}{720}}}"></span></dd></dl> <p>In generale, la risommazione Poisson è utile in quanto una serie che converge lentamente nello spazio diretto può essere trasformato in una serie convergente molto più velocemente nello spazio di Fourier (se prendiamo l'esempio di funzioni gaussiane, una gaussiana varianza grande nello spazio diretto è trasformata in una gaussiana con varianza piccola spazio di Fourier). Questa è l'idea fondamentale alla base della <a href="/w/index.php?title=Sommatoria_di_Ewald&action=edit&redlink=1" class="new" title="Sommatoria di Ewald (la pagina non esiste)">sommatoria di Ewald</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Note">Note</h2></div> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1"><b>^</b></a> <span class="reference-text"><cite class="citation libro" style="font-style:normal"> Mark Pinsky, <span style="font-style:italic;">Introduction to Fourier Analysis and Wavelets</span>, Brooks/Cole, 2001, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/978-0-534-37660-4" title="Speciale:RicercaISBN/978-0-534-37660-4">978-0-534-37660-4</a>.</cite></span> </li> <li id="cite_note-2"><a href="#cite_ref-2"><b>^</b></a> <span class="reference-text"><cite class="citation libro" style="font-style:normal"> Antoni Zygmund, <span style="font-style:italic;">Trigonometric series (2nd ed.)</span>, Cambridge University Press, 1988, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/978-0-521-35885-9" title="Speciale:RicercaISBN/978-0-521-35885-9">978-0-521-35885-9</a>.</cite></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Bibliografia">Bibliografia</h2></div> <ul><li>(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) Matthew R. Watkins, <a rel="nofollow" class="external text" href="http://www.maths.ex.ac.uk/~mwatkins/zeta/physics4.htm">pagina</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20060925091627/http://www.maths.ex.ac.uk/~mwatkins/zeta/physics4.htm">Archiviato</a> il 25 settembre 2006 in <a href="/wiki/Internet_Archive" title="Internet Archive">Internet Archive</a>. sui legami tra teoria dei numeri e la fisica teorica.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Voci_correlate">Voci correlate</h2></div> <ul><li><a href="/wiki/Distribuzione_(matematica)" title="Distribuzione (matematica)">Distribuzione (matematica)</a></li> <li><a href="/w/index.php?title=Formula_di_Selberg&action=edit&redlink=1" class="new" title="Formula di Selberg (la pagina non esiste)">Formula di Selberg</a></li> <li><a href="/wiki/Operatore_di_Laplace-Beltrami" title="Operatore di Laplace-Beltrami">Operatore di Laplace-Beltrami</a></li> <li><a href="/wiki/Serie_di_Fourier" title="Serie di Fourier">Serie di Fourier</a></li> <li><a href="/wiki/Teorema_della_convergenza_dominata" title="Teorema della convergenza dominata">Teorema della convergenza dominata</a></li> <li><a href="/wiki/Trasformata_di_Fourier_discreta" class="mw-redirect" title="Trasformata di Fourier discreta">Trasformata di Fourier discreta</a></li> <li><a href="/wiki/Trasformata_di_Fourier" title="Trasformata di Fourier">Trasformata di Fourier</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Collegamenti_esterni">Collegamenti esterni</h2></div> <ul><li class="mw-empty-elt"></li> <li><cite id="CITEREFMathWorld" class="citation web" style="font-style:normal">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) Eric W. Weisstein, <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/PoissonSumFormula.html"><span style="font-style:italic;">Formula di sommazione di Poisson</span></a>, su <span style="font-style:italic;"><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></span>, Wolfram Research.</cite> <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q387743#P2812" title="Modifica su Wikidata"><img alt="Modifica su Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/10px-Blue_pencil.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/15px-Blue_pencil.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/20px-Blue_pencil.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></li></ul> <div class="noprint" style="width:100%; padding: 3px 0; display: flex; flex-wrap: wrap; row-gap: 4px; column-gap: 8px; box-sizing: border-box;"><div style="flex-grow: 1"><style data-mw-deduplicate="TemplateStyles:r140555418">.mw-parser-output .itwiki-template-occhiello{width:100%;line-height:25px;border:1px solid #CCF;background-color:#F0EEFF;box-sizing:border-box}.mw-parser-output .itwiki-template-occhiello-progetto{background-color:#FAFAFA}@media screen{html.skin-theme-clientpref-night .mw-parser-output .itwiki-template-occhiello{background-color:#202122;border-color:#54595D}html.skin-theme-clientpref-night .mw-parser-output .itwiki-template-occhiello-progetto{background-color:#282929}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .itwiki-template-occhiello{background-color:#202122;border-color:#54595D}html.skin-theme-clientpref-os .mw-parser-output .itwiki-template-occhiello-progetto{background-color:#282929}}</style><div class="itwiki-template-occhiello"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Crystal128-kmplot.svg" class="mw-file-description" title="Matematica"><img alt=" " src="//upload.wikimedia.org/wikipedia/commons/thumb/a/af/Crystal128-kmplot.svg/25px-Crystal128-kmplot.svg.png" decoding="async" width="25" height="25" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/af/Crystal128-kmplot.svg/38px-Crystal128-kmplot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/af/Crystal128-kmplot.svg/50px-Crystal128-kmplot.svg.png 2x" data-file-width="245" data-file-height="244" /></a></span> <b><a href="/wiki/Portale:Matematica" title="Portale:Matematica">Portale Matematica</a></b>: accedi alle voci di Wikipedia che trattano di matematica</div></div></div></div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Estratto da "<a dir="ltr" href="https://it.wikipedia.org/w/index.php?title=Formula_di_sommazione_di_Poisson&oldid=132810333">https://it.wikipedia.org/w/index.php?title=Formula_di_sommazione_di_Poisson&oldid=132810333</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Categoria:Categorie" title="Categoria:Categorie">Categoria</a>: <ul><li><a href="/wiki/Categoria:Analisi_di_Fourier" title="Categoria:Analisi di Fourier">Analisi di Fourier</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Categorie nascoste: <ul><li><a href="/wiki/Categoria:Contestualizzare_fonti_-_matematica" title="Categoria:Contestualizzare fonti - matematica">Contestualizzare fonti - matematica</a></li><li><a href="/wiki/Categoria:Contestualizzare_fonti_-_aprile_2023" title="Categoria:Contestualizzare fonti - aprile 2023">Contestualizzare fonti - aprile 2023</a></li><li><a href="/wiki/Categoria:Template_Webarchive_-_collegamenti_all%27Internet_Archive" title="Categoria:Template Webarchive - collegamenti all'Internet Archive">Template Webarchive - collegamenti all'Internet Archive</a></li><li><a href="/wiki/Categoria:P2812_letta_da_Wikidata" title="Categoria:P2812 letta da Wikidata">P2812 letta da Wikidata</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> Questa pagina è stata modificata per l'ultima volta il 2 apr 2023 alle 08:09.</li> <li id="footer-info-copyright">Il testo è disponibile secondo la <a rel="nofollow" class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.it">licenza Creative Commons Attribuzione-Condividi allo stesso modo</a>; possono applicarsi condizioni ulteriori. 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Formula di sommazione di Poisson: differenze tra le versioni - Wikipedia