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Ortogonální polynomy – Wikipedie
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open-block";block.previousSibling.className+=" open-block";}</script> <div class="mw-content-ltr mw-parser-output" lang="cs" dir="ltr"> <section class="mf-section-0" id="mf-section-0"> <p><b>Posloupnost ortogonálních polynomů</b> je v <a href="https://cs-m-wikipedia-org.translate.goog/wiki/Matematika?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Matematika">matematice</a> rodina <a href="https://cs-m-wikipedia-org.translate.goog/wiki/Polynom?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Polynom">polynomů</a> taková, že jakékoli dva různé polynomy v posloupnosti jsou navzájem <a href="https://cs-m-wikipedia-org.translate.goog/wiki/Ortogonalita?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Ortogonalita">ortogonální</a> v nějakém <a href="https://cs-m-wikipedia-org.translate.goog/wiki/Unit%C3%A1rn%C3%AD_prostor?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Unitární prostor">unitárním prostoru</a>.</p> <p>Nejpoužívanější ortogonální polynomy jsou <a href="https://cs-m-wikipedia-org.translate.goog/w/index.php?title=Klasick%C3%A9_ortogon%C3%A1ln%C3%AD_polynomy&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Klasické ortogonální polynomy (stránka neexistuje)">klasické ortogonální polynomy</a>, ke kterým patří <a href="https://cs-m-wikipedia-org.translate.goog/wiki/Hermitovy_polynomy?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Hermitovy polynomy">Hermitovy polynomy</a>, <a href="https://cs-m-wikipedia-org.translate.goog/wiki/Laguerrovy_polynomy?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Laguerrovy polynomy">Laguerrovy polynomy</a> a <a href="https://cs-m-wikipedia-org.translate.goog/w/index.php?title=Jacobiho_polynomy&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Jacobiho polynomy (stránka neexistuje)">Jacobiho polynomy</a> spolu s jejich speciálními případy <a href="https://cs-m-wikipedia-org.translate.goog/w/index.php?title=Gegenbauerovy_polynomy&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Gegenbauerovy polynomy (stránka neexistuje)">Gegenbauerovými polynomy</a>, <a href="https://cs-m-wikipedia-org.translate.goog/w/index.php?title=%C4%8Ceby%C5%A1evovy_polynomy&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Čebyševovy polynomy (stránka neexistuje)">Čebyševovými polynomy</a> a <a href="https://cs-m-wikipedia-org.translate.goog/wiki/Legendrovy_polynomy?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Legendrovy polynomy">Legendrovými polynomy</a>.</p> <p>Obor ortogonálních polynomů rozvinul na konci 19. století ze studia <a href="https://cs-m-wikipedia-org.translate.goog/wiki/%C5%98et%C4%9Bzov%C3%BD_zlomek?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Řetězový zlomek">řetězových zlomků</a> <a href="https://cs-m-wikipedia-org.translate.goog/wiki/Pafnutij_Lvovi%C4%8D_%C4%8Ceby%C5%A1ev?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Pafnutij Lvovič Čebyšev">Pafnutij Lvovič Čebyšev</a> a rozvíjeli jej <a href="https://cs-m-wikipedia-org.translate.goog/wiki/Andrej_Markov?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Andrej Markov">Andrej Markov</a> a <a href="https://cs-m-wikipedia-org.translate.goog/w/index.php?title=Thomas_Joannes_Stieltjes&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Thomas Joannes Stieltjes (stránka neexistuje)">Thomas Joannes Stieltjes</a>. K dalším matematikům, kteří se zabývali ortogonálními polynomy, patří <a href="https://cs-m-wikipedia-org.translate.goog/w/index.php?title=G%C3%A1bor_Szeg%C5%91&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Gábor Szegő (stránka neexistuje)">Gábor Szegő</a>, <a href="https://cs-m-wikipedia-org.translate.goog/w/index.php?title=Sergej_Natanovi%C4%8D_Bernstein&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Sergej Natanovič Bernstein (stránka neexistuje)">Sergej Natanovič Bernstein</a>, <a href="https://cs-m-wikipedia-org.translate.goog/w/index.php?title=Naum_Ilji%C4%8D_Achiezer&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Naum Iljič Achiezer (stránka neexistuje)">Naum Iljič Achiezer</a>, <a href="https://cs-m-wikipedia-org.translate.goog/w/index.php?title=Arthur_Erd%C3%A9lyi&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Arthur Erdélyi (stránka neexistuje)">Arthur Erdélyi</a>, <a href="https://cs-m-wikipedia-org.translate.goog/w/index.php?title=Jakov_Lazarovi%C4%8D_Geronimus&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Jakov Lazarovič Geronimus (stránka neexistuje)">Jakov Lazarovič Geronimus</a>, <a href="https://cs-m-wikipedia-org.translate.goog/w/index.php?title=Wolfgang_Hahn&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Wolfgang Hahn (stránka neexistuje)">Wolfgang Hahn</a>, <a href="https://cs-m-wikipedia-org.translate.goog/w/index.php?title=Theodore_Seio_Chihara&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Theodore Seio Chihara (stránka neexistuje)">Theodore Seio Chihara</a>, <a href="https://cs-m-wikipedia-org.translate.goog/w/index.php?title=Mourad_Ismail&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Mourad Ismail (stránka neexistuje)">Mourad Ismail</a>, <a href="https://cs-m-wikipedia-org.translate.goog/w/index.php?title=Waleed_Al-Salam&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Waleed Al-Salam (stránka neexistuje)">Waleed Al-Salam</a> a <a href="https://cs-m-wikipedia-org.translate.goog/w/index.php?title=Richard_Askey&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Richard Askey (stránka neexistuje)">Richard Askey</a>.</p> <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"> <input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none"> <div class="toctitle" lang="cs" dir="ltr"> <h2 id="mw-toc-heading">Obsah</h2><span class="toctogglespan"><label class="toctogglelabel" for="toctogglecheckbox"></label></span> </div> <ul> <li class="toclevel-1 tocsection-1"><a href="https://cs-m-wikipedia-org.translate.goog/wiki/Ortogon%C3%A1ln%C3%AD_polynomy?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Definice_pro_p%C5%99%C3%ADpad_jedn%C3%A9_prom%C4%9Bnn%C3%A1_s_re%C3%A1lnou_m%C3%ADrou"><span class="tocnumber">1</span> <span class="toctext">Definice pro případ jedné proměnná s reálnou mírou</span></a> <ul> <li class="toclevel-2 tocsection-2"><a href="https://cs-m-wikipedia-org.translate.goog/wiki/Ortogon%C3%A1ln%C3%AD_polynomy?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#P%C5%99%C3%ADpad_absolutn%C4%9B_spojit%C3%A9_funkce"><span class="tocnumber">1.1</span> <span class="toctext">Případ absolutně spojité funkce</span></a></li> </ul></li> <li class="toclevel-1 tocsection-3"><a href="https://cs-m-wikipedia-org.translate.goog/wiki/Ortogon%C3%A1ln%C3%AD_polynomy?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#P%C5%99%C3%ADklady_ortogon%C3%A1ln%C3%ADch_polynom%C5%AF"><span class="tocnumber">2</span> <span class="toctext">Příklady ortogonálních polynomů</span></a></li> <li class="toclevel-1 tocsection-4"><a href="https://cs-m-wikipedia-org.translate.goog/wiki/Ortogon%C3%A1ln%C3%AD_polynomy?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Vlastnosti"><span class="tocnumber">3</span> <span class="toctext">Vlastnosti</span></a> <ul> <li class="toclevel-2 tocsection-5"><a href="https://cs-m-wikipedia-org.translate.goog/wiki/Ortogon%C3%A1ln%C3%AD_polynomy?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Vztah_k_moment%C5%AFm"><span class="tocnumber">3.1</span> <span class="toctext">Vztah k momentům</span></a></li> <li class="toclevel-2 tocsection-6"><a href="https://cs-m-wikipedia-org.translate.goog/wiki/Ortogon%C3%A1ln%C3%AD_polynomy?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Rekurentn%C3%AD_vztah"><span class="tocnumber">3.2</span> <span class="toctext">Rekurentní vztah</span></a></li> <li class="toclevel-2 tocsection-7"><a href="https://cs-m-wikipedia-org.translate.goog/wiki/Ortogon%C3%A1ln%C3%AD_polynomy?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Christoffel%C5%AFv%E2%80%93Darboux%C5%AFv_vzorec"><span class="tocnumber">3.3</span> <span class="toctext">Christoffelův–Darbouxův vzorec</span></a></li> <li class="toclevel-2 tocsection-8"><a href="https://cs-m-wikipedia-org.translate.goog/wiki/Ortogon%C3%A1ln%C3%AD_polynomy?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Ko%C5%99eny"><span class="tocnumber">3.4</span> <span class="toctext">Kořeny</span></a></li> </ul></li> <li class="toclevel-1 tocsection-9"><a href="https://cs-m-wikipedia-org.translate.goog/wiki/Ortogon%C3%A1ln%C3%AD_polynomy?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#V%C3%ADcerozm%C4%9Brn%C3%A9_ortogon%C3%A1ln%C3%AD_polynomy"><span class="tocnumber">4</span> <span class="toctext">Vícerozměrné ortogonální polynomy</span></a></li> <li class="toclevel-1 tocsection-10"><a href="https://cs-m-wikipedia-org.translate.goog/wiki/Ortogon%C3%A1ln%C3%AD_polynomy?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Odkazy"><span class="tocnumber">5</span> <span class="toctext">Odkazy</span></a> <ul> <li class="toclevel-2 tocsection-11"><a href="https://cs-m-wikipedia-org.translate.goog/wiki/Ortogon%C3%A1ln%C3%AD_polynomy?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Reference"><span class="tocnumber">5.1</span> <span class="toctext">Reference</span></a></li> <li class="toclevel-2 tocsection-12"><a href="https://cs-m-wikipedia-org.translate.goog/wiki/Ortogon%C3%A1ln%C3%AD_polynomy?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Literatura"><span class="tocnumber">5.2</span> <span class="toctext">Literatura</span></a></li> <li class="toclevel-2 tocsection-13"><a href="https://cs-m-wikipedia-org.translate.goog/wiki/Ortogon%C3%A1ln%C3%AD_polynomy?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Souvisej%C3%ADc%C3%AD_%C4%8Dl%C3%A1nky"><span class="tocnumber">5.3</span> <span class="toctext">Související články</span></a></li> <li class="toclevel-2 tocsection-14"><a href="https://cs-m-wikipedia-org.translate.goog/wiki/Ortogon%C3%A1ln%C3%AD_polynomy?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Extern%C3%AD_odkazy"><span class="tocnumber">5.4</span> <span class="toctext">Externí odkazy</span></a></li> </ul></li> </ul> </div> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(1)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="Definice_pro_případ_jedné_proměnná_s_reálnou_mírou"><span id="Definice_pro_p.C5.99.C3.ADpad_jedn.C3.A9_prom.C4.9Bnn.C3.A1_s_re.C3.A1lnou_m.C3.ADrou"></span>Definice pro případ jedné proměnná s reálnou mírou</h2><span class="mw-editsection"> <a role="button" href="https://cs-m-wikipedia-org.translate.goog/w/index.php?title=Ortogon%C3%A1ln%C3%AD_polynomy&action=edit&section=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Editace sekce: Definice pro případ jedné proměnná s reálnou mírou" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>editovat</span> </a> </span> </div> <section class="mf-section-1 collapsible-block" id="mf-section-1"> <p>Je-li dána nějaká neklesající funkce <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 {\mbox{d}}\alpha }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext> d </mtext> </mstyle> </mrow> <mi> α<!-- α --> </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\mbox{d}}\alpha } </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e9b14fa8016fab3a52aeb08f6c8c7570cf88323" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.78ex; height:2.176ex;" alt="{\displaystyle {\mbox{d}}\alpha }"> </noscript><span class="lazy-image-placeholder" style="width: 2.78ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e9b14fa8016fab3a52aeb08f6c8c7570cf88323" data-alt="{\displaystyle {\mbox{d}}\alpha }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> na reálných číslech, můžeme definovat <a href="https://cs-m-wikipedia-org.translate.goog/w/index.php?title=Lebesgue%C5%AFv%E2%80%93Stieltjes%C5%AFv_integr%C3%A1l&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Lebesgueův–Stieltjesův integrál (stránka neexistuje)">Lebesgueův–Stieltjesův integrál</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 \int f(x)\;{\mbox{d}}\alpha (x)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo> ∫<!-- ∫ --> </mo> <mi> f </mi> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <mspace width="thickmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext> d </mtext> </mstyle> </mrow> <mi> α<!-- α --> </mi> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \int f(x)\;{\mbox{d}}\alpha (x)} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/045246ce550d8c8f471762c4f96cf779dd9eac95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:13.563ex; height:5.676ex;" alt="{\displaystyle \int f(x)\;{\mbox{d}}\alpha (x)}"> </noscript><span class="lazy-image-placeholder" style="width: 13.563ex;height: 5.676ex;vertical-align: -2.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/045246ce550d8c8f471762c4f96cf779dd9eac95" data-alt="{\displaystyle \int f(x)\;{\mbox{d}}\alpha (x)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>funkce <i>f</i>. Pokud je tento integrál konečný pro všechny polynomy <i>f</i>, můžeme definovat vnitřní součin dvojice polynomů <i>f</i> a <i>g</i> vzorcem</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 \langle f,g\rangle =\int f(x)g(x)\;{\mbox{d}}\alpha (x).}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false"> ⟨<!-- ⟨ --> </mo> <mi> f </mi> <mo> , </mo> <mi> g </mi> <mo fence="false" stretchy="false"> ⟩<!-- ⟩ --> </mo> <mo> = </mo> <mo> ∫<!-- ∫ --> </mo> <mi> f </mi> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <mi> g </mi> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <mspace width="thickmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext> d </mtext> </mstyle> </mrow> <mi> α<!-- α --> </mi> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \langle f,g\rangle =\int f(x)g(x)\;{\mbox{d}}\alpha (x).} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8df537e3292ad2bd2558714d14a554438e4b0018" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:26.801ex; height:5.676ex;" alt="{\displaystyle \langle f,g\rangle =\int f(x)g(x)\;{\mbox{d}}\alpha (x).}"> </noscript><span class="lazy-image-placeholder" style="width: 26.801ex;height: 5.676ex;vertical-align: -2.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8df537e3292ad2bd2558714d14a554438e4b0018" data-alt="{\displaystyle \langle f,g\rangle =\int f(x)g(x)\;{\mbox{d}}\alpha (x).}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>Tato operace je pozitivně semidefinitní <a href="https://cs-m-wikipedia-org.translate.goog/wiki/Unit%C3%A1rn%C3%AD_prostor?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Unitární prostor">vnitřní součin</a> na <a href="https://cs-m-wikipedia-org.translate.goog/wiki/Vektorov%C3%BD_prostor?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Vektorový prostor">vektorovém prostoru</a> všech polynomů a je pozitivně definitní, pokud funkce α má nekonečný počet bodů růstu. Obvyklým způsobem zavedeme pojem <a href="https://cs-m-wikipedia-org.translate.goog/wiki/Ortogonalita?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Ortogonalita">ortogonality</a>, jmenovitě, že dva polynomy jsou ortogonální, pokud je jejich vnitřní součin nula.</p> <p>Pak posloupnost (<i>P</i><sub><i>n</i></sub>)<sub><i>n</i>=0</sub><sup>∞</sup> ortogonálních polynomů je definována vztahy</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 \deg P_{n}=n~,\quad \langle P_{m},\,P_{n}\rangle =0\quad {\text{for}}\quad m\neq n~.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> deg </mi> <mo> <!-- --> </mo> <msub> <mi> P </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> <mo> = </mo> <mi> n </mi> <mtext> </mtext> <mo> , </mo> <mspace width="1em"></mspace> <mo fence="false" stretchy="false"> ⟨<!-- ⟨ --> </mo> <msub> <mi> P </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> m </mi> </mrow> </msub> <mo> , </mo> <mspace width="thinmathspace"></mspace> <msub> <mi> P </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> <mo fence="false" stretchy="false"> ⟩<!-- ⟩ --> </mo> <mo> = </mo> <mn> 0 </mn> <mspace width="1em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mtext> for </mtext> </mrow> <mspace width="1em"></mspace> <mi> m </mi> <mo> ≠<!-- ≠ --> </mo> <mi> n </mi> <mtext> </mtext> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \deg P_{n}=n~,\quad \langle P_{m},\,P_{n}\rangle =0\quad {\text{for}}\quad m\neq n~.} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa49eb6d6cc7301bf9d20f2648c89521ff1aca13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.577ex; height:2.843ex;" alt="{\displaystyle \deg P_{n}=n~,\quad \langle P_{m},\,P_{n}\rangle =0\quad {\text{for}}\quad m\neq n~.}"> </noscript><span class="lazy-image-placeholder" style="width: 43.577ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa49eb6d6cc7301bf9d20f2648c89521ff1aca13" data-alt="{\displaystyle \deg P_{n}=n~,\quad \langle P_{m},\,P_{n}\rangle =0\quad {\text{for}}\quad m\neq n~.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>Jinými slovy posloupnost získáme z posloupnosti jednočlenů 1, <i>x</i>, <i>x</i><sup>2</sup>, ... <a href="https://cs-m-wikipedia-org.translate.goog/wiki/Gramova%E2%80%93Schmidtova_ortogonalizace?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Gramova–Schmidtova ortogonalizace">Gram–Schmidtovou ortogonalizací</a> vzhledem k tomuto vnitřnímu součinu.</p> <p>Obvykle požadujeme, aby posloupnost byla <a href="https://cs-m-wikipedia-org.translate.goog/wiki/Ortonormalita?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Ortonormalita">ortonormální</a>; zpravidla, aby</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 \langle P_{n},P_{n}\rangle =1~,}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false"> ⟨<!-- ⟨ --> </mo> <msub> <mi> P </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> <mo> , </mo> <msub> <mi> P </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> <mo fence="false" stretchy="false"> ⟩<!-- ⟩ --> </mo> <mo> = </mo> <mn> 1 </mn> <mtext> </mtext> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \langle P_{n},P_{n}\rangle =1~,} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7041342023ddbbe6c3af2bfe858dd01103ac2597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.753ex; height:2.843ex;" alt="{\displaystyle \langle P_{n},P_{n}\rangle =1~,}"> </noscript><span class="lazy-image-placeholder" style="width: 13.753ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7041342023ddbbe6c3af2bfe858dd01103ac2597" data-alt="{\displaystyle \langle P_{n},P_{n}\rangle =1~,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>ale někdy se používají jiné normalizace.</p> <div class="mw-heading mw-heading3"> <h3 id="Případ_absolutně_spojité_funkce"><span id="P.C5.99.C3.ADpad_absolutn.C4.9B_spojit.C3.A9_funkce"></span>Případ absolutně spojité funkce</h3><span class="mw-editsection"> <a role="button" href="https://cs-m-wikipedia-org.translate.goog/w/index.php?title=Ortogon%C3%A1ln%C3%AD_polynomy&action=edit&section=2&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Editace sekce: Případ absolutně spojité funkce" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>editovat</span> </a> </span> </div> <p>Někdy máme</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 {\mbox{d}}\alpha (x)=W(x)\,{\mbox{d}}x}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext> d </mtext> </mstyle> </mrow> <mi> α<!-- α --> </mi> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <mi> W </mi> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext> d </mtext> </mstyle> </mrow> <mi> x </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\mbox{d}}\alpha (x)=W(x)\,{\mbox{d}}x} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01ca697b61d66870438e7c62b2e53a872856e0cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.601ex; height:2.843ex;" alt="{\displaystyle {\mbox{d}}\alpha (x)=W(x)\,{\mbox{d}}x}"> </noscript><span class="lazy-image-placeholder" style="width: 17.601ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01ca697b61d66870438e7c62b2e53a872856e0cb" data-alt="{\displaystyle {\mbox{d}}\alpha (x)=W(x)\,{\mbox{d}}x}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>kde</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 W:\langle x_{1},x_{2}\rangle \to \mathbb {R} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> W </mi> <mo> : </mo> <mo fence="false" stretchy="false"> ⟨<!-- ⟨ --> </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo> , </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> <mo fence="false" stretchy="false"> ⟩<!-- ⟩ --> </mo> <mo stretchy="false"> →<!-- → --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck"> R </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle W:\langle x_{1},x_{2}\rangle \to \mathbb {R} } </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be9fb1e21e1a666fc01bb8692dfe512bd2c3d86" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.276ex; height:2.843ex;" alt="{\displaystyle W:\langle x_{1},x_{2}\rangle \to \mathbb {R} }"> </noscript><span class="lazy-image-placeholder" style="width: 17.276ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be9fb1e21e1a666fc01bb8692dfe512bd2c3d86" data-alt="{\displaystyle W:\langle x_{1},x_{2}\rangle \to \mathbb {R} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>je nezáporná funkce s nosičem na nějakém intervalu <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 \langle x_{1},x_{2}\rangle }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false"> ⟨<!-- ⟨ --> </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo> , </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> <mo fence="false" stretchy="false"> ⟩<!-- ⟩ --> </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \langle x_{1},x_{2}\rangle } </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0e932113e183e99e2bdccb18ca7b2f93abbc867" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.611ex; height:2.843ex;" alt="{\displaystyle \langle x_{1},x_{2}\rangle }"> </noscript><span class="lazy-image-placeholder" style="width: 7.611ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0e932113e183e99e2bdccb18ca7b2f93abbc867" data-alt="{\displaystyle \langle x_{1},x_{2}\rangle }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> reálné osy (přičemž může být i <i>x</i><sub>1</sub> = −∞ a <i>x</i><sub>2</sub> = ∞). Takové <i>W</i> se nazývá <a href="https://cs-m-wikipedia-org.translate.goog/wiki/V%C3%A1hov%C3%A1_funkce?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Váhová funkce">váhová funkce</a>. Pak vnitřní součin popisuje vztah</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 \langle f,g\rangle =\int _{x_{1}}^{x_{2}}f(x)g(x)W(x)\;{\mbox{d}}x.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false"> ⟨<!-- ⟨ --> </mo> <mi> f </mi> <mo> , </mo> <mi> g </mi> <mo fence="false" stretchy="false"> ⟩<!-- ⟩ --> </mo> <mo> = </mo> <msubsup> <mo> ∫<!-- ∫ --> </mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> </mrow> </msubsup> <mi> f </mi> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <mi> g </mi> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <mi> W </mi> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <mspace width="thickmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext> d </mtext> </mstyle> </mrow> <mi> x </mi> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \langle f,g\rangle =\int _{x_{1}}^{x_{2}}f(x)g(x)W(x)\;{\mbox{d}}x.} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b74d61fe5a78240909db8b09024663c94a741050" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:31.353ex; height:6.176ex;" alt="{\displaystyle \langle f,g\rangle =\int _{x_{1}}^{x_{2}}f(x)g(x)W(x)\;{\mbox{d}}x.}"> </noscript><span class="lazy-image-placeholder" style="width: 31.353ex;height: 6.176ex;vertical-align: -2.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b74d61fe5a78240909db8b09024663c94a741050" data-alt="{\displaystyle \langle f,g\rangle =\int _{x_{1}}^{x_{2}}f(x)g(x)W(x)\;{\mbox{d}}x.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>Existuje však mnoho příkladů ortogonálních polynomů, kde míra dα(<i>x</i>) má body s nenulovou mírou, ve kterých je funkce α nespojitá, takže <a href="https://cs-m-wikipedia-org.translate.goog/wiki/V%C3%A1hov%C3%A1_funkce?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Váhová funkce">váhovou funkci</a> <i>W</i> nelze definovat jako výše.</p> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(2)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="Příklady_ortogonálních_polynomů"><span id="P.C5.99.C3.ADklady_ortogon.C3.A1ln.C3.ADch_polynom.C5.AF"></span>Příklady ortogonálních polynomů</h2><span class="mw-editsection"> <a role="button" href="https://cs-m-wikipedia-org.translate.goog/w/index.php?title=Ortogon%C3%A1ln%C3%AD_polynomy&action=edit&section=3&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Editace sekce: Příklady ortogonálních polynomů" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>editovat</span> </a> </span> </div> <section class="mf-section-2 collapsible-block" id="mf-section-2"> <p>Nejpoužívanější ortogonální polynomy jsou ortogonální pro míru s nosičem na nějakém reálném intervalu. Patří k nim:</p> <ul> <li>Klasické ortogonální polynomy (<a href="https://cs-m-wikipedia-org.translate.goog/w/index.php?title=Jacobiho_polynomy&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Jacobiho polynomy (stránka neexistuje)">Jacobiho polynomy</a>, <a href="https://cs-m-wikipedia-org.translate.goog/wiki/Laguerrovy_polynomy?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Laguerrovy polynomy">Laguerrovy polynomy</a>, <a href="https://cs-m-wikipedia-org.translate.goog/wiki/Hermitovy_polynomy?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Hermitovy polynomy">Hermitovy polynomy</a> a jejich speciální případy <a href="https://cs-m-wikipedia-org.translate.goog/w/index.php?title=Gegenbauer_polynomy&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Gegenbauer polynomy (stránka neexistuje)">Gegenbauer polynomy</a>, <a href="https://cs-m-wikipedia-org.translate.goog/w/index.php?title=%C4%8Ceby%C5%A1evovy_polynomy&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Čebyševovy polynomy (stránka neexistuje)">Čebyševovy polynomy</a> a <a href="https://cs-m-wikipedia-org.translate.goog/wiki/Legendrovy_polynomy?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Legendrovy polynomy">Legendrovy polynomy</a>).</li> <li><a href="https://cs-m-wikipedia-org.translate.goog/w/index.php?title=Wilsonovy_polynomy&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Wilsonovy polynomy (stránka neexistuje)">Wilsonovy polynomy</a> zobecňující Jacobiho polynomy. Mnoho ortogonálních polynomů sem patří jako speciální případy, například <a href="https://cs-m-wikipedia-org.translate.goog/w/index.php?title=Meixnerovy%E2%80%93Pollaczekovy_polynomy&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Meixnerovy–Pollaczekovy polynomy (stránka neexistuje)">Meixnerovy–Pollaczekovy polynomy</a>, <a href="https://cs-m-wikipedia-org.translate.goog/w/index.php?title=Spojit%C3%A9_Hahnovy_polynomy&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Spojité Hahnovy polynomy (stránka neexistuje)">spojité Hahnovy polynomy</a>, <a href="https://cs-m-wikipedia-org.translate.goog/w/index.php?title=Spojit%C3%A9_du%C3%A1ln%C3%AD_Hahnovy_polynomy&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Spojité duální Hahnovy polynomy (stránka neexistuje)">spojité duální Hahnovy polynomy</a> a klasické polynomy popsané <a href="https://cs-m-wikipedia-org.translate.goog/w/index.php?title=Askeyovo_sch%C3%A9ma&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Askeyovo schéma (stránka neexistuje)">Askeyovým schématem</a>.</li> <li><a href="https://cs-m-wikipedia-org.translate.goog/w/index.php?title=Askeyovy%E2%80%93Wilsonovy_polynomy&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Askeyovy–Wilsonovy polynomy (stránka neexistuje)">Askeyovy–Wilsonovy polynomy</a> zavádějí do Wilsonových polynomů zvláštní parametr <i>q</i>.</li> </ul> <p><a href="https://cs-m-wikipedia-org.translate.goog/w/index.php?title=Diskr%C3%A9tn%C3%AD_ortogon%C3%A1ln%C3%AD_polynomy&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Diskrétní ortogonální polynomy (stránka neexistuje)">Diskrétní ortogonální polynomy</a> jsou ortogonální vzhledem k nějaké diskrétní míře. Míra má někdy konečný nosič; v tomto případě není rodina ortogonálních polynomů nekonečnou posloupností, ale konečnou. <a href="https://cs-m-wikipedia-org.translate.goog/w/index.php?title=Racahovy_polynomy&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Racahovy polynomy (stránka neexistuje)">Racahovy polynomy</a> jsou příkladem diskrétních ortogonálních polynomů a jako speciální případy zahrnují <a href="https://cs-m-wikipedia-org.translate.goog/w/index.php?title=Hahnovy_polynomy&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Hahnovy polynomy (stránka neexistuje)">Hahnovy polynomy</a> a <a href="https://cs-m-wikipedia-org.translate.goog/w/index.php?title=Du%C3%A1ln%C3%AD_Hahnovy_polynomy&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Duální Hahnovy polynomy (stránka neexistuje)">duální Hahnovy polynomy</a>, které zase zahrnují jako speciální případy <a href="https://cs-m-wikipedia-org.translate.goog/w/index.php?title=Meixnerovy_polynomy&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Meixnerovy polynomy (stránka neexistuje)">Meixnerovy polynomy</a>, <a href="https://cs-m-wikipedia-org.translate.goog/w/index.php?title=Krav%C4%8Dukovy_polynomy&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Kravčukovy polynomy (stránka neexistuje)">Kravčukovy polynomy</a> a <a href="https://cs-m-wikipedia-org.translate.goog/w/index.php?title=Charlierovy_polynomy&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Charlierovy polynomy (stránka neexistuje)">Charlierovy polynomy</a>.</p> <p>„Proseté“ ortogonální polynomy (<a href="https://cs-m-wikipedia-org.translate.goog/wiki/Angli%C4%8Dtina?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Angličtina">anglicky</a> <span class="cizojazycne" lang="en" title="angličtina"><i>Sieved orthogonal polynomials</i></span>) jsou ortogonální polynomy, jejichž rekurentní vztah je upraven použitím rekurentního vztahu z jiné skupiny polynomů.</p> <p>Můžeme také uvažovat ortogonální polynomy pro nějaké křivky v komplexní rovině. Nejdůležitějším případem (jiným než reálné intervaly) je, když křivkou je jednotková kružnice, což dává <a href="https://cs-m-wikipedia-org.translate.goog/w/index.php?title=Ortogon%C3%A1ln%C3%AD_polynomy_na_jednotkov%C3%A9_kru%C5%BEnici&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Ortogonální polynomy na jednotkové kružnici (stránka neexistuje)">ortogonální polynomy na jednotkové kružnici</a>, jako například <a href="https://cs-m-wikipedia-org.translate.goog/w/index.php?title=Rogersovy%E2%80%93Szeg%C5%91ovy_polynomy&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Rogersovy–Szegőovy polynomy (stránka neexistuje)">Rogersovy–Szegőovy polynomy</a>.</p> <p>Existují určité rodiny ortogonálních polynomů, které jsou ortogonální na rovinné oblasti jako například na trojúhelnících nebo kruzích. Někdy mohou být zapsány pomocí členů Jacobiho polynomů. Například <a href="https://cs-m-wikipedia-org.translate.goog/w/index.php?title=Zernikeovy_polynomy&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Zernikeovy polynomy (stránka neexistuje)">Zernikeovy polynomy</a> jsou ortogonální na jednotkovém kruhu.</p> <p>Výhoda ortogonality mezi různými řády <a href="https://cs-m-wikipedia-org.translate.goog/wiki/Hermitovy_polynomy?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Hermitovy polynomy">Hermitových polynomů</a> je aplikována na strukturu Zobecněného multiplexování s frekvenčním dělením (<a href="https://cs-m-wikipedia-org.translate.goog/wiki/Angli%C4%8Dtina?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Angličtina">anglicky</a> <span class="cizojazycne" lang="en" title="angličtina"><i>Generalized frequency division multiplexing</i>, <i>GFDM</i></span>). V každé buňce mřížky čas-frekvence může být přenášen více než jeden symbol.<sup id="cite_ref-1" class="reference"><a href="https://cs-m-wikipedia-org.translate.goog/wiki/Ortogon%C3%A1ln%C3%AD_polynomy?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></p> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(3)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="Vlastnosti">Vlastnosti</h2><span class="mw-editsection"> <a role="button" href="https://cs-m-wikipedia-org.translate.goog/w/index.php?title=Ortogon%C3%A1ln%C3%AD_polynomy&action=edit&section=4&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Editace sekce: Vlastnosti" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>editovat</span> </a> </span> </div> <section class="mf-section-3 collapsible-block" id="mf-section-3"> <p>Ortogonální polynomy jedné proměnné definované vztahem nezáporné míry na reálné ose mají následující vlastnosti.</p> <div class="mw-heading mw-heading3"> <h3 id="Vztah_k_momentům"><span id="Vztah_k_moment.C5.AFm"></span>Vztah k momentům</h3><span class="mw-editsection"> <a role="button" href="https://cs-m-wikipedia-org.translate.goog/w/index.php?title=Ortogon%C3%A1ln%C3%AD_polynomy&action=edit&section=5&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Editace sekce: Vztah k momentům" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>editovat</span> </a> </span> </div> <p>Ortogonální polynomy <i>P</i><sub><i>n</i></sub> mohou být vyjádřeny pomocí <a href="https://cs-m-wikipedia-org.translate.goog/wiki/Obecn%C3%BD_moment?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Obecný moment">momentů</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 m_{n}=\int x^{n}\,{\mbox{d}}\alpha (x)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> <mo> = </mo> <mo> ∫<!-- ∫ --> </mo> <msup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext> d </mtext> </mstyle> </mrow> <mi> α<!-- α --> </mi> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle m_{n}=\int x^{n}\,{\mbox{d}}\alpha (x)} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb51cb07d15ac619edd60c05f36e946c1f51cf9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:17.792ex; height:5.676ex;" alt="{\displaystyle m_{n}=\int x^{n}\,{\mbox{d}}\alpha (x)}"> </noscript><span class="lazy-image-placeholder" style="width: 17.792ex;height: 5.676ex;vertical-align: -2.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb51cb07d15ac619edd60c05f36e946c1f51cf9e" data-alt="{\displaystyle m_{n}=\int x^{n}\,{\mbox{d}}\alpha (x)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>takto:</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 P_{n}(x)=c_{n}\,\det {\begin{pmatrix}m_{0}&m_{1}&m_{2}&\cdots &m_{n}\\m_{1}&m_{2}&m_{3}&\cdots &m_{n+1}\\&&\vdots &&\vdots \\m_{n-1}&m_{n}&m_{n+1}&\cdots &m_{2n-1}\\1&x&x^{2}&\cdots &x^{n}\end{pmatrix}}~,}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> P </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <msub> <mi> c </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> <mspace width="thinmathspace"></mspace> <mo movablelimits="true" form="prefix"> det </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo> ( </mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> </mtd> <mtd> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> </mtd> <mtd> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> </mtd> <mtd> <mo> ⋯<!-- ⋯ --> </mo> </mtd> <mtd> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> </mtd> <mtd> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> </mtd> <mtd> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msub> </mtd> <mtd> <mo> ⋯<!-- ⋯ --> </mo> </mtd> <mtd> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd></mtd> <mtd></mtd> <mtd> <mo> ⋮<!-- ⋮ --> </mo> </mtd> <mtd></mtd> <mtd> <mo> ⋮<!-- ⋮ --> </mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> <mo> −<!-- − --> </mo> <mn> 1 </mn> </mrow> </msub> </mtd> <mtd> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> </mtd> <mtd> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> </mtd> <mtd> <mo> ⋯<!-- ⋯ --> </mo> </mtd> <mtd> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> <mi> n </mi> <mo> −<!-- − --> </mo> <mn> 1 </mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn> 1 </mn> </mtd> <mtd> <mi> x </mi> </mtd> <mtd> <msup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mtd> <mtd> <mo> ⋯<!-- ⋯ --> </mo> </mtd> <mtd> <msup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msup> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> </mrow> <mtext> </mtext> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle P_{n}(x)=c_{n}\,\det {\begin{pmatrix}m_{0}&m_{1}&m_{2}&\cdots &m_{n}\\m_{1}&m_{2}&m_{3}&\cdots &m_{n+1}\\&&\vdots &&\vdots \\m_{n-1}&m_{n}&m_{n+1}&\cdots &m_{2n-1}\\1&x&x^{2}&\cdots &x^{n}\end{pmatrix}}~,} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a5f45d671ba58b796fa89489eb349accf769524" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.005ex; width:53.783ex; height:17.176ex;" alt="{\displaystyle P_{n}(x)=c_{n}\,\det {\begin{pmatrix}m_{0}&m_{1}&m_{2}&\cdots &m_{n}\\m_{1}&m_{2}&m_{3}&\cdots &m_{n+1}\\&&\vdots &&\vdots \\m_{n-1}&m_{n}&m_{n+1}&\cdots &m_{2n-1}\\1&x&x^{2}&\cdots &x^{n}\end{pmatrix}}~,}"> </noscript><span class="lazy-image-placeholder" style="width: 53.783ex;height: 17.176ex;vertical-align: -8.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a5f45d671ba58b796fa89489eb349accf769524" data-alt="{\displaystyle P_{n}(x)=c_{n}\,\det {\begin{pmatrix}m_{0}&m_{1}&m_{2}&\cdots &m_{n}\\m_{1}&m_{2}&m_{3}&\cdots &m_{n+1}\\&&\vdots &&\vdots \\m_{n-1}&m_{n}&m_{n+1}&\cdots &m_{2n-1}\\1&x&x^{2}&\cdots &x^{n}\end{pmatrix}}~,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>kde konstanty <i>c</i><sub><i>n</i></sub> jsou libovolné (závisejí na normalizaci polynomů <i>P</i><sub><i>n</i></sub>).</p> <div class="mw-heading mw-heading3"> <h3 id="Rekurentní_vztah"><span id="Rekurentn.C3.AD_vztah"></span>Rekurentní vztah</h3><span class="mw-editsection"> <a role="button" href="https://cs-m-wikipedia-org.translate.goog/w/index.php?title=Ortogon%C3%A1ln%C3%AD_polynomy&action=edit&section=6&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Editace sekce: Rekurentní vztah" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>editovat</span> </a> </span> </div> <p>Polynomy <i>P</i><sub><i>n</i></sub> vyhovují rekurentnímu vztahu tvaru</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 P_{n}(x)=(A_{n}x+B_{n})P_{n-1}(x)+C_{n}P_{n-2}(x)~.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> P </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <mo stretchy="false"> ( </mo> <msub> <mi> A </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> <mi> x </mi> <mo> + </mo> <msub> <mi> B </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <msub> <mi> P </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> <mo> −<!-- − --> </mo> <mn> 1 </mn> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <mo> + </mo> <msub> <mi> C </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> <msub> <mi> P </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> <mo> −<!-- − --> </mo> <mn> 2 </mn> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <mtext> </mtext> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle P_{n}(x)=(A_{n}x+B_{n})P_{n-1}(x)+C_{n}P_{n-2}(x)~.} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32cc8cc8ca460e9eefe5d721f95ff39bb28fde01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.72ex; height:2.843ex;" alt="{\displaystyle P_{n}(x)=(A_{n}x+B_{n})P_{n-1}(x)+C_{n}P_{n-2}(x)~.}"> </noscript><span class="lazy-image-placeholder" style="width: 43.72ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32cc8cc8ca460e9eefe5d721f95ff39bb28fde01" data-alt="{\displaystyle P_{n}(x)=(A_{n}x+B_{n})P_{n-1}(x)+C_{n}P_{n-2}(x)~.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>Opačný výsledek popisuje <a href="https://cs-m-wikipedia-org.translate.goog/w/index.php?title=Favardova_v%C4%9Bta&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Favardova věta (stránka neexistuje)">Favardova věta</a>.</p> <div class="mw-heading mw-heading3"> <h3 id="Christoffelův–Darbouxův_vzorec"><span id="Christoffel.C5.AFv.E2.80.93Darboux.C5.AFv_vzorec"></span>Christoffelův–Darbouxův vzorec</h3><span class="mw-editsection"> <a role="button" href="https://cs-m-wikipedia-org.translate.goog/w/index.php?title=Ortogon%C3%A1ln%C3%AD_polynomy&action=edit&section=7&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Editace sekce: Christoffelův–Darbouxův vzorec" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>editovat</span> </a> </span> </div> <div class="uvodni-upozorneni hatnote"> Související informace naleznete také v článku <a href="https://cs-m-wikipedia-org.translate.goog/w/index.php?title=Christoffel%C5%AFv%E2%80%93Darboux%C5%AFv_vzorec&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Christoffelův–Darbouxův vzorec (stránka neexistuje)">Christoffelův–Darbouxův vzorec</a>. </div> <div class="mw-heading mw-heading3"> <h3 id="Kořeny"><span id="Ko.C5.99eny"></span>Kořeny</h3><span class="mw-editsection"> <a role="button" href="https://cs-m-wikipedia-org.translate.goog/w/index.php?title=Ortogon%C3%A1ln%C3%AD_polynomy&action=edit&section=8&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Editace sekce: Kořeny" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>editovat</span> </a> </span> </div> <p>Pokud míra <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 {\mbox{d}}\alpha }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext> d </mtext> </mstyle> </mrow> <mi> α<!-- α --> </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\mbox{d}}\alpha } </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e9b14fa8016fab3a52aeb08f6c8c7570cf88323" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.78ex; height:2.176ex;" alt="{\displaystyle {\mbox{d}}\alpha }"> </noscript><span class="lazy-image-placeholder" style="width: 2.78ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e9b14fa8016fab3a52aeb08f6c8c7570cf88323" data-alt="{\displaystyle {\mbox{d}}\alpha }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> je podporovaná na intervalu <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 \langle a,b\rangle }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false"> ⟨<!-- ⟨ --> </mo> <mi> a </mi> <mo> , </mo> <mi> b </mi> <mo fence="false" stretchy="false"> ⟩<!-- ⟩ --> </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \langle a,b\rangle } </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f9c6fc93d2e4195b610bc8a9ff366be4485387" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.071ex; height:2.843ex;" alt="{\displaystyle \langle a,b\rangle }"> </noscript><span class="lazy-image-placeholder" style="width: 5.071ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f9c6fc93d2e4195b610bc8a9ff366be4485387" data-alt="{\displaystyle \langle a,b\rangle }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>, všechny kořeny polynomu <i>P</i><sub><i>n</i></sub> leží v <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 \langle a,b\rangle }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false"> ⟨<!-- ⟨ --> </mo> <mi> a </mi> <mo> , </mo> <mi> b </mi> <mo fence="false" stretchy="false"> ⟩<!-- ⟩ --> </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \langle a,b\rangle } </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f9c6fc93d2e4195b610bc8a9ff366be4485387" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.071ex; height:2.843ex;" alt="{\displaystyle \langle a,b\rangle }"> </noscript><span class="lazy-image-placeholder" style="width: 5.071ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f9c6fc93d2e4195b610bc8a9ff366be4485387" data-alt="{\displaystyle \langle a,b\rangle }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>. Navíc mají kořeny následující prokládací vlastnost:, pokud je <i>m</i> < <i>n</i>, existuje kořen polynomu <i>P</i><sub><i>n</i></sub> mezi jakýmikoli dvěma kořeny polynomu <i>P</i><sub><i>m</i></sub>.</p> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(4)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="Vícerozměrné_ortogonální_polynomy"><span id="V.C3.ADcerozm.C4.9Brn.C3.A9_ortogon.C3.A1ln.C3.AD_polynomy"></span>Vícerozměrné ortogonální polynomy</h2><span class="mw-editsection"> <a role="button" href="https://cs-m-wikipedia-org.translate.goog/w/index.php?title=Ortogon%C3%A1ln%C3%AD_polynomy&action=edit&section=9&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Editace sekce: Vícerozměrné ortogonální polynomy" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>editovat</span> </a> </span> </div> <section class="mf-section-4 collapsible-block" id="mf-section-4"> <p><a href="https://cs-m-wikipedia-org.translate.goog/w/index.php?title=Macdonaldovy_polynomy&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Macdonaldovy polynomy (stránka neexistuje)">Macdonaldovy polynomy</a> jsou ortogonální polynomy několika proměnných, v závislosti na volbě affinního kořenového systému. Patří k nim mnoho jiných rodin ortogonálních polynomů více proměnných jako speciální případy, mj. <a href="https://cs-m-wikipedia-org.translate.goog/w/index.php?title=Jackovy_polynomy&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Jackovy polynomy (stránka neexistuje)">Jackovy polynomy</a>, <a href="https://cs-m-wikipedia-org.translate.goog/w/index.php?title=Hallovy%E2%80%93Littlewoodovy_polynomy&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Hallovy–Littlewoodovy polynomy (stránka neexistuje)">Hallovy–Littlewoodovy polynomy</a>, <a href="https://cs-m-wikipedia-org.translate.goog/w/index.php?title=Heckmanovy%E2%80%93Opdamovy_polynomy&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Heckmanovy–Opdamovy polynomy (stránka neexistuje)">Heckmanovy–Opdamovy polynomy</a> a <a href="https://cs-m-wikipedia-org.translate.goog/w/index.php?title=Koornwinderovy_polynomy&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Koornwinderovy polynomy (stránka neexistuje)">Koornwinderovy polynomy</a>. <a href="https://cs-m-wikipedia-org.translate.goog/w/index.php?title=Askeyovy%E2%80%93Wilsonovy_polynomy&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Askeyovy–Wilsonovy polynomy (stránka neexistuje)">Askeyovy–Wilsonovy polynomy</a> jsou speciálním případem Macdonaldových polynomů pro určitý neredukovaný kořenový systém hodnosti 1.</p> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(5)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="Odkazy">Odkazy</h2><span class="mw-editsection"> <a role="button" href="https://cs-m-wikipedia-org.translate.goog/w/index.php?title=Ortogon%C3%A1ln%C3%AD_polynomy&action=edit&section=10&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Editace sekce: Odkazy" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>editovat</span> </a> </span> </div> <section class="mf-section-5 collapsible-block" id="mf-section-5"> <div class="mw-heading mw-heading3"> <h3 id="Reference">Reference</h3><span class="mw-editsection"> <a role="button" href="https://cs-m-wikipedia-org.translate.goog/w/index.php?title=Ortogon%C3%A1ln%C3%AD_polynomy&action=edit&section=11&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Editace sekce: Reference" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>editovat</span> </a> </span> </div> <p><span class="plainlinks"><i>V tomto článku byl použit <a href="https://cs-m-wikipedia-org.translate.goog/wiki/Wikipedie:WikiProjekt_P%C5%99eklad/Rady?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Wikipedie:WikiProjekt Překlad/Rady">překlad</a> textu z článku <a class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://en.wikipedia.org/wiki/Orthogonal_polynomials?oldid%3D999238216"><span class="cizojazycne" lang="en" title="angličtina">Orthogonal polynomials</span></a> na anglické Wikipedii.</i></span></p> <div class="mw-references-wrap"> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="https://cs-m-wikipedia-org.translate.goog/wiki/Ortogon%C3%A1ln%C3%AD_polynomy?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-1">↑</a></span> <span class="reference-text"><cite style="font-style:normal;" id="CITEREFCatakDurak-Ata">CATAK, E.; DURAK-ATA, L. An efficient transceiver design for superimposed waveforms with orthogonal polynomials. <i>IEEE International Black Sea Conference on Communications and Networking (BlackSeaCom)</i>. 2017, s. 1–5. <span style="white-space:nowrap"><a href="https://cs-m-wikipedia-org.translate.goog/wiki/International_Standard_Book_Number?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="International Standard Book Number">ISBN</a> <a href="https://cs-m-wikipedia-org.translate.goog/wiki/Speci%C3%A1ln%C3%AD:Zdroje_knih/978-1-5090-5049-9?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Speciální:Zdroje knih/978-1-5090-5049-9"><span class="ISBN">978-1-5090-5049-9</span></a></span>. <a href="https://cs-m-wikipedia-org.translate.goog/wiki/Digital_object_identifier?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Digital object identifier">doi</a>:<a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://doi.org/10.1109%252FBlackSeaCom.2017.8277657">10.1109/BlackSeaCom.2017.8277657</a>. <a href="https://cs-m-wikipedia-org.translate.goog/wiki/Semantic_Scholar?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Semantic Scholar">S2CID</a> <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://api.semanticscholar.org/CorpusID:22592277"><span class="S2CID">22592277</span></a>.</cite><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rfr_id=info:sid/cs.wikipedia.org:templatecitaceperiodika&rft.jtitle=IEEE+International+Black+Sea+Conference+on+Communications+and+Networking+%28BlackSeaCom%29&rft_id=info:doi/10.1109%2FBlackSeaCom.2017.8277657&rft.atitle=An+efficient+transceiver+design+for+superimposed+waveforms+with+orthogonal+polynomials&rft.date=2017&rft.pages=1%E2%80%935&rft.aulast=Catak&rft.aufirst=E.&rft.au=Durak-Ata%2C+L."><span style="display:none"> </span></span></span></li> </ol> </div> <div class="mw-heading mw-heading3"> <h3 id="Literatura">Literatura</h3><span class="mw-editsection"> <a role="button" href="https://cs-m-wikipedia-org.translate.goog/w/index.php?title=Ortogon%C3%A1ln%C3%AD_polynomy&action=edit&section=12&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Editace sekce: Literatura" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>editovat</span> </a> </span> </div> <ul> <li><cite class="book" style="font-style:normal;" id="CITEREFAbramowitzStegun1972">ABRAMOWITZ, Milton; STEGUN, Irena Ann, 1972. <i>Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables</i>. 9. vyd. Washington D.C.; New York: Dover. <span style="white-space:nowrap"><a href="https://cs-m-wikipedia-org.translate.goog/wiki/International_Standard_Book_Number?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="International Standard Book Number">ISBN</a> <a href="https://cs-m-wikipedia-org.translate.goog/wiki/Speci%C3%A1ln%C3%AD:Zdroje_knih/978-0-486-61272-0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Speciální:Zdroje knih/978-0-486-61272-0"><span class="ISBN">978-0-486-61272-0</span></a></span>. Kapitola 22.</cite><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/cs.wikipedia.org:templatecitacemonografie&rft.btitle=Handbook+of+Mathematical+Functions+with+Formulas%2C+Graphs%2C+and+Mathematical+Tables&rft.isbn=978-0-486-61272-0&rft.aulast=Abramowitz&rft.aufirst=Milton&rft.au=Stegun%2C+Irena+Ann&rft.atitle=22&rft.place=Washington+D.C.%3B+New+York&rft.pub=Dover&rft.date=1972&rft.edition=9"><span style="display:none"> </span></span></li> <li><cite class="book" style="font-style:normal;" id="CITEREFChihara1978">CHIHARA, Theodore Seio, 1978. <i>An Introduction to Orthogonal Polynomials</i>. [s.l.]: Gordon and Breach, New York. <span style="white-space:nowrap"><a href="https://cs-m-wikipedia-org.translate.goog/wiki/International_Standard_Book_Number?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="International Standard Book Number">ISBN</a> <a href="https://cs-m-wikipedia-org.translate.goog/wiki/Speci%C3%A1ln%C3%AD:Zdroje_knih/0-677-04150-0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Speciální:Zdroje knih/0-677-04150-0"><span class="ISBN">0-677-04150-0</span></a></span>.</cite><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/cs.wikipedia.org:templatecitacemonografie&rft.btitle=An+Introduction+to+Orthogonal+Polynomials&rft.isbn=0-677-04150-0&rft.aulast=Chihara&rft.aufirst=Theodore+Seio&rft.pub=Gordon+and+Breach%2C+New+York&rft.date=1978"><span style="display:none"> </span></span></li> <li><cite style="font-style:normal;" id="CITEREFChihara2001">CHIHARA, Theodore Seio, 2001. 45 years of orthogonal polynomials: a view from the wings. <i>Journal of Computational and Applied Mathematics</i>. Roč. 133, čís. 1, s. 13–21. <a href="https://cs-m-wikipedia-org.translate.goog/wiki/International_Standard_Serial_Number?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="International Standard Serial Number">ISSN</a> <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://worldcat.org/issn/0377-0427">0377-0427</a>. <a href="https://cs-m-wikipedia-org.translate.goog/wiki/Digital_object_identifier?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Digital object identifier">doi</a>:<a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://doi.org/10.1016%252FS0377-0427%252800%252900632-4">10.1016/S0377-0427(00)00632-4</a>. <a href="https://cs-m-wikipedia-org.translate.goog/wiki/Bibcode?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Bibcode">Bibcode</a> <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=http://adsabs.harvard.edu/abs/2001JCoAM.133...13C">2001JCoAM.133...13C</a>.</cite><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rfr_id=info:sid/cs.wikipedia.org:templatecitaceperiodika&rft.jtitle=Journal+of+Computational+and+Applied+Mathematics&rft_id=info:doi/10.1016%2FS0377-0427%2800%2900632-4&rft.atitle=45+years+of+orthogonal+polynomials%3A+a+view+from+the+wings&rft.date=2001&rft.volume=133&rft.issue=1&rft.pages=13%E2%80%9321&rft.issn=0377-0427&rft.aulast=Chihara&rft.aufirst=Theodore+Seio"><span style="display:none"> </span></span></li> <li><cite style="font-style:normal;" id="CITEREFFoncannonFoncannonPekonen2008">FONCANNON, J. J.; FONCANNON, J. J.; PEKONEN, Osmo, 2008. Review of <i>Classical and quantum orthogonal polynomials in one variable</i> by Mourad Ismail. <i>The Mathematical Intelligencer</i>. Springer New York. Roč. 30, s. 54–60. <a href="https://cs-m-wikipedia-org.translate.goog/wiki/International_Standard_Serial_Number?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="International Standard Serial Number">ISSN</a> <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://worldcat.org/issn/0343-6993">0343-6993</a>. <a href="https://cs-m-wikipedia-org.translate.goog/wiki/Digital_object_identifier?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Digital object identifier">doi</a>:<a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://doi.org/10.1007%252FBF02985757">10.1007/BF02985757</a>. <a href="https://cs-m-wikipedia-org.translate.goog/wiki/Semantic_Scholar?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Semantic Scholar">S2CID</a> <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://api.semanticscholar.org/CorpusID:118133026"><span class="S2CID">118133026</span></a>.</cite><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rfr_id=info:sid/cs.wikipedia.org:templatecitaceperiodika&rft.jtitle=The+Mathematical+Intelligencer&rft_id=info:doi/10.1007%2FBF02985757&rft.atitle=Review+of+%27%27Classical+and+quantum+orthogonal+polynomials+in+one+variable%27%27+by+Mourad+Ismail&rft.date=2008&rft.volume=30&rft.pages=54%E2%80%9360&rft.issn=0343-6993&rft.aulast=Foncannon&rft.aufirst=J.+J.&rft.au=Foncannon%2C+J.+J.&rft.au=Pekonen%2C+Osmo&rft.pub=Springer+New+York"><span style="display:none"> </span></span></li> <li><cite class="book" style="font-style:normal;" id="CITEREFIsmail2005">ISMAIL, Mourad E. H., 2005. <i>Classical and Quantum Orthogonal Polynomials in One Variable</i>. Cambridge: Cambridge Univ. Press. <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=http://www.cambridge.org/us/catalogue/catalogue.asp?isbn%3D9780521782012">Dostupné online</a>. <span style="white-space:nowrap"><a href="https://cs-m-wikipedia-org.translate.goog/wiki/International_Standard_Book_Number?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="International Standard Book Number">ISBN</a> <a href="https://cs-m-wikipedia-org.translate.goog/wiki/Speci%C3%A1ln%C3%AD:Zdroje_knih/0-521-78201-5?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Speciální:Zdroje knih/0-521-78201-5"><span class="ISBN">0-521-78201-5</span></a></span>.</cite><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/cs.wikipedia.org:templatecitacemonografie&rft.btitle=Classical+and+Quantum+Orthogonal+Polynomials+in+One+Variable&rft_id=http%3A%2F%2Fwww.cambridge.org%2Fus%2Fcatalogue%2Fcatalogue.asp%3Fisbn%3D9780521782012&rft.isbn=0-521-78201-5&rft.aulast=Ismail&rft.aufirst=Mourad+E.+H.&rft.place=Cambridge&rft.pub=Cambridge+Univ.+Press&rft.date=2005"><span style="display:none"> </span></span></li> <li><cite class="book" style="font-style:normal;" id="CITEREFJackson2004">JACKSON, Dunham, 2004. <i>Fourier Series and Orthogonal Polynomials</i>. New York: Dover. <span style="white-space:nowrap"><a href="https://cs-m-wikipedia-org.translate.goog/wiki/International_Standard_Book_Number?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="International Standard Book Number">ISBN</a> <a href="https://cs-m-wikipedia-org.translate.goog/wiki/Speci%C3%A1ln%C3%AD:Zdroje_knih/0-486-43808-2?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Speciální:Zdroje knih/0-486-43808-2"><span class="ISBN">0-486-43808-2</span></a></span>.</cite><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/cs.wikipedia.org:templatecitacemonografie&rft.btitle=Fourier+Series+and+Orthogonal+Polynomials&rft.isbn=0-486-43808-2&rft.aulast=Jackson&rft.aufirst=Dunham&rft.place=New+York&rft.pub=Dover&rft.date=2004"><span style="display:none"> </span></span></li> <li><cite class="book" style="font-style:normal;">KOORNWINDER, Tom H.; WONG, Roderick S. C.; KOEKOEK, Roelof; SWARTTOUW, René F. <i>Orthogonal Polynomials</i>. Příprava vydání Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W.. [s.l.]: Cambridge University Press <span style="white-space:nowrap"><a href="https://cs-m-wikipedia-org.translate.goog/wiki/International_Standard_Book_Number?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="International Standard Book Number">ISBN</a> <a href="https://cs-m-wikipedia-org.translate.goog/wiki/Speci%C3%A1ln%C3%AD:Zdroje_knih/978-0-521-19225-5?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Speciální:Zdroje knih/978-0-521-19225-5"><span class="ISBN">978-0-521-19225-5</span></a></span>. p/o070340.</cite><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/cs.wikipedia.org:templatecitacemonografie&rft.btitle=Orthogonal+Polynomials&rft.isbn=978-0-521-19225-5&rft.aulast=Koornwinder&rft.aufirst=Tom+H.&rft.au=Wong%2C+Roderick+S.+C.&rft.au=Koekoek%2C+Roelof&rft.pub=Cambridge+University+Press"><span style="display:none"> </span></span></li> <li><cite class="book" style="font-style:normal;" id="CITEREFSzegő1939">SZEGŐ, Gábor, 1939. <i>Orthogonal Polynomials</i>. [s.l.]: American Mathematical Society. (Colloquium Publications). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://books.google.com/books?id%3D3hcW8HBh7gsC">Dostupné online</a>. <span style="white-space:nowrap"><a href="https://cs-m-wikipedia-org.translate.goog/wiki/International_Standard_Book_Number?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="International Standard Book Number">ISBN</a> <a href="https://cs-m-wikipedia-org.translate.goog/wiki/Speci%C3%A1ln%C3%AD:Zdroje_knih/978-0-8218-1023-1?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Speciální:Zdroje knih/978-0-8218-1023-1"><span class="ISBN">978-0-8218-1023-1</span></a></span>.</cite><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/cs.wikipedia.org:templatecitacemonografie&rft.btitle=Orthogonal+Polynomials&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D3hcW8HBh7gsC&rft.isbn=978-0-8218-1023-1&rft.aulast=Szeg%C5%91&rft.aufirst=G%C3%A1bor&rft.pub=American+Mathematical+Society&rft.date=1939&rft.series=Colloquium+Publications"><span style="display:none"> </span></span></li> <li><cite style="font-style:normal;">SIRCAR, P.; PACHORI, R.B.; KUMAR, R. Analysis of rhythms of EEG signals using orthogonal polynomial approximation. In: <i>ACM International Conference on Convergence and Hybrid Information Technology</i>. Daejeon, South Korea: [s.n.], 27.–29. srpna 2009. S. 176–180.</cite></li> <li><cite style="font-style:normal;" id="CITEREFTotik2005">TOTIK, Vilmos, 2005. Orthogonal Polynomials. <i>Surveys in Approximation Theory</i>. Roč. 1, s. 70–125. <a href="https://cs-m-wikipedia-org.translate.goog/wiki/ArXiv?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="ArXiv">arXiv</a> <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=http://arxiv.org/abs/math.CA%252F0512424">math.CA/0512424</a>.</cite><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rfr_id=info:sid/cs.wikipedia.org:templatecitaceperiodika&rft.jtitle=Surveys+in+Approximation+Theory&rft.atitle=Orthogonal+Polynomials&rft.date=2005&rft.volume=1&rft.pages=70%E2%80%93125&rft.aulast=Totik&rft.aufirst=Vilmos"><span style="display:none"> </span></span></li> <li><cite style="font-style:normal;">Chuan-Tsung Chan; MIRONOV, A.; MOROZOV, A.; SLEPTSOV, A. <i>Reviews in Mathematical Physics</i>. Roč. 2018, čís. 6. <a href="https://cs-m-wikipedia-org.translate.goog/wiki/Digital_object_identifier?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Digital object identifier">doi</a>:<a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://doi.org/10.1142%252FS0129055X18400056">10.1142/S0129055X18400056</a>.</cite><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rfr_id=info:sid/cs.wikipedia.org:templatecitaceperiodika&rft.jtitle=Reviews+in+Mathematical+Physics&rft_id=info:doi/10.1142%2FS0129055X18400056&rft.volume=2018&rft.issue=6&rft.au=Chuan-Tsung+Chan&rft.au=Mironov%2C+A.&rft.au=Morozov%2C+A."><span style="display:none"> </span></span></li> </ul> <div class="mw-heading mw-heading3"> <h3 id="Související_články"><span id="Souvisej.C3.ADc.C3.AD_.C4.8Dl.C3.A1nky"></span>Související články</h3><span class="mw-editsection"> <a role="button" 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href="https://cs-m-wikipedia-org.translate.goog/w/index.php?title=Favardova_v%C4%9Bta&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Favardova věta (stránka neexistuje)">Favardova věta</a></li> <li><a href="https://cs-m-wikipedia-org.translate.goog/w/index.php?title=Binomick%C3%BD_typ&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Binomický typ (stránka neexistuje)">Posloupnosti polynomů binomického typu</a></li> <li><a href="https://cs-m-wikipedia-org.translate.goog/w/index.php?title=Biortogon%C3%A1ln%C3%AD_polynomy&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Biortogonální polynomy (stránka neexistuje)">Biortogonální polynomy</a></li> <li><a href="https://cs-m-wikipedia-org.translate.goog/w/index.php?title=Zobecn%C4%9Bn%C3%A9_Fourierovy_%C5%99ady&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Zobecněné Fourierovy řady (stránka neexistuje)">Zobecněné Fourierovy řady</a></li> <li><a href="https://cs-m-wikipedia-org.translate.goog/w/index.php?title=Sekund%C3%A1rn%C3%AD_m%C3%ADra&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Sekundární míra (stránka neexistuje)">Sekundární míra</a></li> <li><a href="https://cs-m-wikipedia-org.translate.goog/w/index.php?title=Shefferova_posloupnost&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Shefferova posloupnost (stránka neexistuje)">Shefferova posloupnost</a></li> <li><a href="https://cs-m-wikipedia-org.translate.goog/w/index.php?title=Sturmova%E2%80%93Liouvilleova_teorie&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Sturmova–Liouvilleova teorie (stránka neexistuje)">Sturmova–Liouvilleova teorie</a></li> <li><a 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href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ar.wikipedia.org/wiki/%25D9%2585%25D8%25AA%25D8%25B9%25D8%25AF%25D8%25AF%25D8%25A7%25D8%25AA_%25D8%25A7%25D9%2584%25D8%25AD%25D8%25AF%25D9%2588%25D8%25AF_%25D9%2585%25D8%25AA%25D8%25B9%25D8%25A7%25D9%2585%25D8%25AF%25D8%25A9" title="متعددات الحدود متعامدة – arabština" lang="ar" hreflang="ar" data-title="متعددات الحدود متعامدة" data-language-autonym="العربية" data-language-local-name="arabština" class="interlanguage-link-target"><span>العربية</span></a></li> <li class="interlanguage-link interwiki-az mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://az.wikipedia.org/wiki/Ortoqonal_%25C3%25A7oxh%25C9%2599dlil%25C9%2599r" title="Ortoqonal çoxhədlilər – ázerbájdžánština" lang="az" hreflang="az" data-title="Ortoqonal çoxhədlilər" data-language-autonym="Azərbaycanca" data-language-local-name="ázerbájdžánština" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li> <li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ca.wikipedia.org/wiki/Polinomis_ortogonals" title="Polinomis ortogonals – katalánština" lang="ca" hreflang="ca" data-title="Polinomis ortogonals" data-language-autonym="Català" data-language-local-name="katalánština" class="interlanguage-link-target"><span>Català</span></a></li> <li class="interlanguage-link interwiki-de mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://de.wikipedia.org/wiki/Orthogonale_Polynome" title="Orthogonale Polynome – němčina" lang="de" hreflang="de" data-title="Orthogonale Polynome" data-language-autonym="Deutsch" data-language-local-name="němčina" class="interlanguage-link-target"><span>Deutsch</span></a></li> <li class="interlanguage-link interwiki-en mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://en.wikipedia.org/wiki/Orthogonal_polynomials" title="Orthogonal polynomials – angličtina" lang="en" hreflang="en" data-title="Orthogonal polynomials" data-language-autonym="English" data-language-local-name="angličtina" class="interlanguage-link-target"><span>English</span></a></li> <li class="interlanguage-link interwiki-es mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://es.wikipedia.org/wiki/Polinomios_ortogonales" title="Polinomios ortogonales – španělština" lang="es" hreflang="es" data-title="Polinomios ortogonales" data-language-autonym="Español" data-language-local-name="španělština" class="interlanguage-link-target"><span>Español</span></a></li> <li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://fa.wikipedia.org/wiki/%25DA%2586%25D9%2586%25D8%25AF%25D8%25AC%25D9%2585%25D9%2584%25D9%2587%25E2%2580%258C%25D8%25A7%25DB%258C%25E2%2580%258C%25D9%2587%25D8%25A7%25DB%258C_%25D9%2585%25D8%25AA%25D8%25B9%25D8%25A7%25D9%2585%25D8%25AF" title="چندجملهایهای متعامد – perština" lang="fa" hreflang="fa" data-title="چندجملهایهای متعامد" data-language-autonym="فارسی" data-language-local-name="perština" class="interlanguage-link-target"><span>فارسی</span></a></li> <li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://fi.wikipedia.org/wiki/Ortogonaaliset_polynomit" title="Ortogonaaliset polynomit – finština" lang="fi" hreflang="fi" data-title="Ortogonaaliset polynomit" data-language-autonym="Suomi" data-language-local-name="finština" class="interlanguage-link-target"><span>Suomi</span></a></li> <li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://fr.wikipedia.org/wiki/Suite_de_polyn%25C3%25B4mes_orthogonaux" title="Suite de polynômes orthogonaux – francouzština" lang="fr" hreflang="fr" data-title="Suite de polynômes orthogonaux" data-language-autonym="Français" data-language-local-name="francouzština" class="interlanguage-link-target"><span>Français</span></a></li> <li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://hi.wikipedia.org/wiki/%25E0%25A4%25B2%25E0%25A4%25BE%25E0%25A4%2582%25E0%25A4%25AC%25E0%25A4%25BF%25E0%25A4%2595_%25E0%25A4%25AC%25E0%25A4%25B9%25E0%25A5%2581%25E0%25A4%25AA%25E0%25A4%25A6" title="लांबिक बहुपद – hindština" lang="hi" hreflang="hi" data-title="लांबिक बहुपद" data-language-autonym="हिन्दी" data-language-local-name="hindština" class="interlanguage-link-target"><span>हिन्दी</span></a></li> <li class="interlanguage-link interwiki-it mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://it.wikipedia.org/wiki/Polinomi_ortogonali" title="Polinomi ortogonali – italština" lang="it" hreflang="it" data-title="Polinomi ortogonali" data-language-autonym="Italiano" data-language-local-name="italština" class="interlanguage-link-target"><span>Italiano</span></a></li> <li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ja.wikipedia.org/wiki/%25E7%259B%25B4%25E4%25BA%25A4%25E5%25A4%259A%25E9%25A0%2585%25E5%25BC%258F" title="直交多項式 – japonština" lang="ja" hreflang="ja" data-title="直交多項式" data-language-autonym="日本語" data-language-local-name="japonština" class="interlanguage-link-target"><span>日本語</span></a></li> <li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://nl.wikipedia.org/wiki/Orthogonale_polynomen" title="Orthogonale polynomen – nizozemština" lang="nl" hreflang="nl" data-title="Orthogonale polynomen" data-language-autonym="Nederlands" data-language-local-name="nizozemština" class="interlanguage-link-target"><span>Nederlands</span></a></li> <li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://pl.wikipedia.org/wiki/Wielomiany_ortogonalne" title="Wielomiany ortogonalne – polština" lang="pl" hreflang="pl" data-title="Wielomiany ortogonalne" data-language-autonym="Polski" data-language-local-name="polština" class="interlanguage-link-target"><span>Polski</span></a></li> <li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ro.wikipedia.org/wiki/Polinoame_ortogonale" title="Polinoame ortogonale – rumunština" lang="ro" hreflang="ro" data-title="Polinoame ortogonale" data-language-autonym="Română" data-language-local-name="rumunština" class="interlanguage-link-target"><span>Română</span></a></li> <li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ru.wikipedia.org/wiki/%25D0%259E%25D1%2580%25D1%2582%25D0%25BE%25D0%25B3%25D0%25BE%25D0%25BD%25D0%25B0%25D0%25BB%25D1%258C%25D0%25BD%25D1%258B%25D0%25B5_%25D0%25BC%25D0%25BD%25D0%25BE%25D0%25B3%25D0%25BE%25D1%2587%25D0%25BB%25D0%25B5%25D0%25BD%25D1%258B" title="Ортогональные многочлены – ruština" lang="ru" hreflang="ru" data-title="Ортогональные многочлены" data-language-autonym="Русский" data-language-local-name="ruština" class="interlanguage-link-target"><span>Русский</span></a></li> <li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://sl.wikipedia.org/wiki/Ortogonalni_polinomi" title="Ortogonalni polinomi – slovinština" lang="sl" hreflang="sl" data-title="Ortogonalni polinomi" data-language-autonym="Slovenščina" data-language-local-name="slovinština" class="interlanguage-link-target"><span>Slovenščina</span></a></li> <li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://sv.wikipedia.org/wiki/Ortogonala_polynom" title="Ortogonala polynom – švédština" lang="sv" hreflang="sv" data-title="Ortogonala polynom" data-language-autonym="Svenska" data-language-local-name="švédština" class="interlanguage-link-target"><span>Svenska</span></a></li> <li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://tg.wikipedia.org/wiki/%25D0%2591%25D0%25B8%25D1%2581%25D1%2591%25D1%2580%25D1%258A%25D1%2583%25D0%25B7%25D0%25B2%25D0%25B0%25D2%25B3%25D0%25BE%25D0%25B8_%25D0%25BE%25D1%2580%25D1%2582%25D0%25BE%25D0%25B3%25D0%25BE%25D0%25BD%25D0%25B0%25D0%25BB%25D3%25A3" title="Бисёръузваҳои ортогоналӣ – tádžičtina" lang="tg" hreflang="tg" data-title="Бисёръузваҳои ортогоналӣ" data-language-autonym="Тоҷикӣ" data-language-local-name="tádžičtina" class="interlanguage-link-target"><span>Тоҷикӣ</span></a></li> <li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://uk.wikipedia.org/wiki/%25D0%259E%25D1%2580%25D1%2582%25D0%25BE%25D0%25B3%25D0%25BE%25D0%25BD%25D0%25B0%25D0%25BB%25D1%258C%25D0%25BD%25D1%2596_%25D0%25BF%25D0%25BE%25D0%25BB%25D1%2596%25D0%25BD%25D0%25BE%25D0%25BC%25D0%25B8" title="Ортогональні поліноми – ukrajinština" lang="uk" hreflang="uk" data-title="Ортогональні поліноми" data-language-autonym="Українська" data-language-local-name="ukrajinština" class="interlanguage-link-target"><span>Українська</span></a></li> <li class="interlanguage-link interwiki-vec mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://vec.wikipedia.org/wiki/Po%25C5%2582inomi_ortogona%25C5%2582i" title="Połinomi ortogonałi – benátština" lang="vec" hreflang="vec" data-title="Połinomi ortogonałi" data-language-autonym="Vèneto" data-language-local-name="benátština" class="interlanguage-link-target"><span>Vèneto</span></a></li> <li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://zh.wikipedia.org/wiki/%25E6%25AD%25A3%25E4%25BA%25A4%25E5%25A4%259A%25E9%25A0%2585%25E5%25BC%258F" title="正交多項式 – čínština" lang="zh" hreflang="zh" data-title="正交多項式" data-language-autonym="中文" data-language-local-name="čínština" class="interlanguage-link-target"><span>中文</span></a></li> </ul> </section> </div> <div class="minerva-footer-logo"> <img src="/static/images/mobile/copyright/wikipedia-wordmark-cs.svg" alt="Wikipedie" width="120" height="19" style="width: 7.5em; height: 1.1875em;"> </div> <ul id="footer-info" class="footer-info hlist hlist-separated"> <li id="footer-info-lastmod">Stránka byla naposledy editována 1. 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