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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Volkenborn-Integral</span></h1>
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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"><p>Das <b>Volkenborn-Integral</b> ist ein <a href="Integralrechnung" title="Integralrechnung">Integralbegriff</a> für <a href="Funktion_(Mathematik)" title="Funktion (Mathematik)">Funktionen</a> auf den <a href="P-adische_Zahl" title="P-adische Zahl">p-adischen Zahlen</a>.
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<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>
<p>Sei
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon \mathbb {Z} _{p}\rightarrow \mathbb {C} _{p}}">
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<mi>f</mi>
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<annotation encoding="application/x-tex">{\displaystyle f\colon \mathbb {Z} _{p}\rightarrow \mathbb {C} _{p}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1feafbaa15c1c4953419a0cbcce152523f3400b8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.273ex; height:2.843ex;" alt="{\displaystyle f\colon \mathbb {Z} _{p}\rightarrow \mathbb {C} _{p}}" loading="lazy"></span></dd></dl>
<p>eine lokal-analytische Funktion von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} _{p}}">
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<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} _{p}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dbc1df7227ef11fe88dccd2dae3adc7bbdeae5f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.609ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} _{p}}" loading="lazy"></span>, dem Ring der <a href="P-adische_Zahl" title="P-adische Zahl">p-adischen</a> ganzen Zahlen, in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} _{p}}">
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<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} _{p}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b6f9e7692267c8a29ed4d848c3421eee929c23c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.737ex; height:2.843ex;" alt="{\displaystyle \mathbb {C} _{p}}" loading="lazy"></span>, die <a href="Vervollst%C3%A4ndigung_(metrischer_Raum)" class="mw-redirect" title="Vervollständigung (metrischer Raum)">Vervollständigung</a> des <a href="Algebraischer_Abschluss" title="Algebraischer Abschluss">algebraischen Abschlusses</a> von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} _{p}}">
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<annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} _{p}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/35f44bc6894c682710705f3ea74f33042e0acc3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.867ex; height:2.843ex;" alt="{\displaystyle \mathbb {Q} _{p}}" loading="lazy"></span>, dem Körper der <span class="mwe-math-element mwe-math-element-inline"><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}">
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<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>p</mi>
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<annotation encoding="application/x-tex">{\displaystyle p}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"></span>-adischen Zahlen (eine Funktion heißt lokal-analytisch, wenn es um jeden Punkt eine Kreisscheibe gibt, innerhalb derer sich die Funktion in eine Potenzreihe entwickeln lässt). Das Volkenborn-Integral von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">
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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> ist dann definiert durch
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<dl><dd><span class="mwe-math-element mwe-math-element-inline"><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 _{\mathbb {Z} _{p}}f(x)\,{\rm {d}}x=\lim _{n\to \infty }{\frac {1}{p^{n}}}\sum _{x=0}^{p^{n}-1}f(x).}">
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<annotation encoding="application/x-tex">{\displaystyle \int _{\mathbb {Z} _{p}}f(x)\,{\rm {d}}x=\lim _{n\to \infty }{\frac {1}{p^{n}}}\sum _{x=0}^{p^{n}-1}f(x).}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/52acb904ab766977aabb9bd84165fd3a8c6bb818.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:31.984ex; height:7.509ex;" alt="{\displaystyle \int _{\mathbb {Z} _{p}}f(x)\,{\rm {d}}x=\lim _{n\to \infty }{\frac {1}{p^{n}}}\sum _{x=0}^{p^{n}-1}f(x).}" loading="lazy"></span></dd></dl>
<div class="mw-heading mw-heading2"><h2 id="Entstehung">Entstehung</h2></div>
<p>Die Idee der Integration von p-adischen Funktionen hatten zunächst F. Thomas und <a href="Fran%C3%A7ois_Bruhat" title="François Bruhat">F. Bruhat</a>. Die Definition ihres <a href="Translationsinvarianz" class="mw-redirect" title="Translationsinvarianz">translationsinvarianten</a> p-adischen Integrals erwies sich aber als zu restriktiv für analytische und zahlentheoretische Zwecke.
</p><p><a href="Arnt_Volkenborn" title="Arnt Volkenborn">Arnt Volkenborn</a> entwickelte in seiner Dissertation an der <a href="Universit%C3%A4t_zu_K%C3%B6ln" title="Universität zu Köln">Universität zu Köln</a> 1971 das später nach ihm benannte verallgemeinerte <i><span class="mwe-math-element mwe-math-element-inline"><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}">
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<annotation encoding="application/x-tex">{\displaystyle p}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"></span>-adische Integral</i>.
Mit dem <i>Volkenborn-Integral</i> werden alle lokal-analytischen Funktionen, wie die <a href="Laurent-Reihe" title="Laurent-Reihe">Laurent-Reihen</a>, integrierbar. Anwendung erfährt das <i>Volkenborn-Integral</i> bei der Berechnung der sogenannten verallgemeinerten <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>p</mi>
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<annotation encoding="application/x-tex">{\displaystyle p}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"></span>-<a href="Bernoulli-Zahl" title="Bernoulli-Zahl">Bernoulli-Zahlen</a> und weiteren <span class="mwe-math-element mwe-math-element-inline"><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}">
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<mi>p</mi>
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<annotation encoding="application/x-tex">{\displaystyle p}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"></span>-adischen Funktionen.
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<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>
<ul><li>Arnt Volkenborn: <i>Ein p-adisches Integral und seine Anwendungen I.</i> In: <i>Manuscripta Mathematica.</i> Bd. 7, Nr. 4, 1972, <span class="-print"><a href="Internationale_Standardnummer_f%C3%BCr_fortlaufende_Sammelwerke" title="Internationale Standardnummer für fortlaufende Sammelwerke">ISSN</a>&nbsp;<span style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://zdb-katalog.de/list.xhtml?t=iss%3D%220025-2611%22&amp;key=cql">0025-2611</a></span></span>, S. 341–373. <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<span class="uri-handle" style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://doi.org/10.1007/BF01644073">10.1007/BF01644073</a></span></li>
<li>Arnt Volkenborn: <i>Ein p-adisches Integral und seine Anwendungen II.</i> In: <i>Manuscripta Mathematica.</i> Bd. 12, Nr. 1, 1974, <span class="-print"><a href="Internationale_Standardnummer_f%C3%BCr_fortlaufende_Sammelwerke" title="Internationale Standardnummer für fortlaufende Sammelwerke">ISSN</a>&nbsp;<span style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://zdb-katalog.de/list.xhtml?t=iss%3D%220025-2611%22&amp;key=cql">0025-2611</a></span></span>, S. 17–46. <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<span class="uri-handle" style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://doi.org/10.1007/BF01166232">10.1007/BF01166232</a></span></li>
<li><a href="Alain_M._Robert" class="mw-redirect" title="Alain M. Robert">Alain M. Robert</a>: <i>A Course on p-adic Analysis</i> (= <i>Graduate Texts in Mathematics.</i> Bd. 198). Springer, New York u. a. 2000, ISBN 0-387-98669-3, S. 263–279.</li>
<li>Min-Soo Kim, Jin-Woo Son: <i>Analytic Properties of the q-Volkenborn Integral on the Ring of p-Adic Integers.</i> In: <i>Bulletin of the Korean Mathematical Society.</i> Bd. 44, Nr. 1, 2007, <span class="-print"><a href="Internationale_Standardnummer_f%C3%BCr_fortlaufende_Sammelwerke" title="Internationale Standardnummer für fortlaufende Sammelwerke">ISSN</a>&nbsp;<span style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://zdb-katalog.de/list.xhtml?t=iss%3D%221015-8634%22&amp;key=cql">1015-8634</a></span></span>, S. 1–12, <a rel="nofollow" class="external text" href="https://www.mathnet.or.kr/mathnet/kms_content.php?no=378936">online</a>.</li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
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