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class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="사이트"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="목차" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">목차</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">사이드바로 이동</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">숨기기</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">처음 위치</div> </a> </li> <li id="toc-예시" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#예시"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>예시</span> </div> </a> <button aria-controls="toc-예시-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>예시 하위섹션 토글하기</span> </button> <ul id="toc-예시-sublist" class="vector-toc-list"> <li id="toc-다항식_기저" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#다항식_기저"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>다항식 기저</span> </div> </a> <ul id="toc-다항식_기저-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-푸리에_기저" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#푸리에_기저"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>푸리에 기저</span> </div> </a> <ul id="toc-푸리에_기저-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-같이_보기" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#같이_보기"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>같이 보기</span> </div> </a> <ul id="toc-같이_보기-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="목차" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="목차 토글" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">목차 토글</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">기저 함수</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="다른 언어로 문서를 방문합니다. 11개 언어로 읽을 수 있습니다" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-11" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">11개 언어</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Basis_function" title="Basis function – 영어" lang="en" hreflang="en" data-title="Basis function" data-language-autonym="English" data-language-local-name="영어" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Baza_funkcio" title="Baza funkcio – 에스페란토어" lang="eo" hreflang="eo" data-title="Baza funkcio" data-language-autonym="Esperanto" data-language-local-name="에스페란토어" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Funci%C3%B3n_base" title="Función base – 스페인어" lang="es" hreflang="es" data-title="Función base" data-language-autonym="Español" data-language-local-name="스페인어" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AA%D8%A7%D8%A8%D8%B9_%D8%A8%D9%86%DB%8C%D8%A7%D8%AF%DB%8C" title="تابع بنیادی – 페르시아어" lang="fa" hreflang="fa" data-title="تابع بنیادی" data-language-autonym="فارسی" data-language-local-name="페르시아어" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Fonction_de_base" title="Fonction de base – 프랑스어" lang="fr" hreflang="fr" data-title="Fonction de base" data-language-autonym="Français" data-language-local-name="프랑스어" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A4%D7%A2%D7%95%D7%9C%D7%AA_%D7%91%D7%A1%D7%99%D7%A1" title="פעולת בסיס – 히브리어" lang="he" hreflang="he" data-title="פעולת בסיס" data-language-autonym="עברית" data-language-local-name="히브리어" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%9F%BA%E5%BA%95%E9%96%A2%E6%95%B0" title="基底関数 – 일본어" lang="ja" hreflang="ja" data-title="基底関数" data-language-autonym="日本語" data-language-local-name="일본어" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Basisfunksjon" title="Basisfunksjon – 노르웨이어(니노르스크)" lang="nn" hreflang="nn" data-title="Basisfunksjon" data-language-autonym="Norsk nynorsk" data-language-local-name="노르웨이어(니노르스크)" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%91%D0%B0%D0%B7%D0%B8%D1%81%D0%BD%D0%B0%D1%8F_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F" title="Базисная функция – 러시아어" lang="ru" hreflang="ru" data-title="Базисная функция" data-language-autonym="Русский" data-language-local-name="러시아어" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%91%D0%B0%D0%B7%D0%B8%D1%81%D0%BD%D0%B0_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D1%96%D1%8F" title="Базисна функція – 우크라이나어" lang="uk" hreflang="uk" data-title="Базисна функція" data-language-autonym="Українська" data-language-local-name="우크라이나어" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%9F%BA%E5%87%BD%E6%95%B8" title="基函數 – 중국어" lang="zh" hreflang="zh" data-title="基函數" data-language-autonym="中文" data-language-local-name="중국어" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q2621825#sitelinks-wikipedia" title="언어 간 링크 편집" class="wbc-editpage">링크 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항목을 가리키는 링크 [g]" accesskey="g"><span>위키데이터 항목</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="페이지 도구"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="보이기"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">보이기</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">사이드바로 이동</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">숨기기</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">위키백과, 우리 모두의 백과사전.</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="ko" dir="ltr"><p><span class="nowrap"></span> <b>기저 함수</b>(basis function) 또는 <b>바탕 함수</b>란 <a href="/wiki/%ED%95%A8%EC%88%98_%EA%B3%B5%EA%B0%84" title="함수 공간">함수 공간</a>의 <a href="/wiki/%EA%B8%B0%EC%A0%80" class="mw-disambig" title="기저">기저</a>인 함수를 말한다. 모든 <a href="/wiki/%EB%B2%A1%ED%84%B0_%EA%B3%B5%EA%B0%84" title="벡터 공간">벡터 공간</a>의 함수들을 <a href="/wiki/%EA%B8%B0%EC%A0%80_%EB%B2%A1%ED%84%B0" class="mw-redirect" title="기저 벡터">기저 벡터</a>의 <a href="/wiki/%EC%84%A0%ED%98%95%EA%B2%B0%ED%95%A9" class="mw-redirect" title="선형결합">선형결합</a>으로 표시할 수 있듯이 모든 연속함수들은 기저 함수들의 <a href="/wiki/%EC%84%A0%ED%98%95%EA%B2%B0%ED%95%A9" class="mw-redirect" title="선형결합">선형결합</a>으로 표시할 수 있다. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="예시"><span id=".EC.98.88.EC.8B.9C"></span>예시</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EA%B8%B0%EC%A0%80_%ED%95%A8%EC%88%98&action=edit&section=1" title="부분 편집: 예시"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="다항식_기저"><span id=".EB.8B.A4.ED.95.AD.EC.8B.9D_.EA.B8.B0.EC.A0.80"></span>다항식 기저</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EA%B8%B0%EC%A0%80_%ED%95%A8%EC%88%98&action=edit&section=2" title="부분 편집: 다항식 기저"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>실계수 2차 다항식에서 {1, <i>t</i>, <i>t</i><sup>2</sup>} 은 기저 함수이다. 모든 실계수 2차 다항식은 <i>a</i>1+<i>bt</i>+<i>ct</i><sup>2</sup>의 꼴로 표현되므로, 기저함수 1, <i>t</i>, and <i>t</i><sup>2</sup>의 선형결합으로 표시된다. {(<i>t</i>−1)(<i>t</i>−2)/2, −<i>t</i>(<i>t</i>−2), <i>t</i>(<i>t</i>−1)/2}의 세 함수들은 이차 다항식 다른 기저함수가 되며 <a href="/wiki/%EB%9D%BC%EA%B7%B8%EB%9E%91%EC%A3%BC_%EB%8B%A4%ED%95%AD%EC%8B%9D" title="라그랑주 다항식">라그랑주 기저</a>라고 불린다. <a href="/wiki/%EC%B2%B4%EB%B9%84%EC%87%BC%ED%94%84_%EB%8B%A4%ED%95%AD%EC%8B%9D" title="체비쇼프 다항식">체비쇼프 다항식</a>의 처음 세 항도 2차 다항식의 다른 기저함수가 된다. </p> <div class="mw-heading mw-heading3"><h3 id="푸리에_기저"><span id=".ED.91.B8.EB.A6.AC.EC.97.90_.EA.B8.B0.EC.A0.80"></span>푸리에 기저</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EA%B8%B0%EC%A0%80_%ED%95%A8%EC%88%98&action=edit&section=3" title="부분 편집: 푸리에 기저"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>사인과 코사인은 <a href="/w/index.php?title=%EC%A0%9C%EA%B3%B1%ED%95%B4%EC%84%9C_%EC%A0%81%EB%B6%84%EA%B0%80%EB%8A%A5%ED%95%9C_%ED%95%A8%EC%88%98&action=edit&redlink=1" class="new" title="제곱해서 적분가능한 함수 (없는 문서)">제곱해서 적분가능한 함수</a>들의 (정규직교) <a href="/wiki/%EC%83%A4%EC%9A%B0%EB%8D%B0%EB%A5%B4_%EA%B8%B0%EC%A0%80" title="샤우데르 기저">샤우데르 기저</a>가 된다. 예를 들어 다음 함수들의 모음은 </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 \{{\sqrt {2}}\sin(2\pi nx)\;|\;n\in \mathbb {N} \}\cup \{{\sqrt {2}}\cos(2\pi nx)\;|\;n\in \mathbb {N} \}\cup \{1\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mi>n</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mspace width="thickmathspace" /> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo fence="false" stretchy="false">}</mo> <mo>∪</mo> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mi>n</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mspace width="thickmathspace" /> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo fence="false" stretchy="false">}</mo> <mo>∪</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{{\sqrt {2}}\sin(2\pi nx)\;|\;n\in \mathbb {N} \}\cup \{{\sqrt {2}}\cos(2\pi nx)\;|\;n\in \mathbb {N} \}\cup \{1\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9947de43e43fde2b67f2cd663c392cd65a760f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:55.997ex; height:3.176ex;" alt="{\displaystyle \{{\sqrt {2}}\sin(2\pi nx)\;|\;n\in \mathbb {N} \}\cup \{{\sqrt {2}}\cos(2\pi nx)\;|\;n\in \mathbb {N} \}\cup \{1\}}"></span></dd></dl> <p><a href="/wiki/Lp_%EA%B3%B5%EA%B0%84" class="mw-redirect" title="Lp 공간">L<sup>2</sup>(0,1)</a>의 기저를 이룬다. </p> <div class="mw-heading mw-heading2"><h2 id="같이_보기"><span id=".EA.B0.99.EC.9D.B4_.EB.B3.B4.EA.B8.B0"></span>같이 보기</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EA%B8%B0%EC%A0%80_%ED%95%A8%EC%88%98&action=edit&section=4" title="부분 편집: 같이 보기"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/%EA%B8%B0%EC%A0%80_(%EC%84%A0%ED%98%95%EB%8C%80%EC%88%98%ED%95%99)" title="기저 (선형대수학)">기저 (선형대수학)</a></li> <li><a href="/wiki/%EB%B0%94%ED%83%95_%ED%95%A8%EC%88%98_%EC%A7%91%ED%95%A9" title="바탕 함수 집합">바탕 함수 집합</a></li> <li><a href="/wiki/%EC%8C%8D%EB%8C%80%EA%B8%B0%EC%A0%80" class="mw-redirect" title="쌍대기저">쌍대기저</a></li> <li><a href="/w/index.php?title=%EC%8C%8D%EC%A7%81%EA%B5%90%EA%B3%84&action=edit&redlink=1" class="new" title="쌍직교계 (없는 문서)">쌍직교계</a></li> <li><a href="/wiki/%EC%A0%95%EA%B7%9C_%EC%A7%81%EA%B5%90_%EA%B8%B0%EC%A0%80" title="정규 직교 기저">정규 직교 기저</a></li> <li><a href="/wiki/%EB%82%B4%EC%A0%81%EA%B3%B5%EA%B0%84" class="mw-redirect" title="내적공간">내적공간</a></li> <li><a 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