Density of sampling and interpolation in reproducing kernel Hilbert spaces

Führ Hartmut; Gr?chenig Karlheinz; Haimi Antti; Klotz Andreas; Romero José Luis

摘要

Abstract > We derive necessary density conditions for sampling and for interpolation in general reproducing kernel Hilbert spaces satisfying some natural conditions on the geometry of the space and the reproducing kernel. If the volume of shells is small compared to the volume of balls (weak annular decay property) and if the kernel possesses some off‐diagonal decay or even some weaker form of localization, then there exists a critical density <mat:math xmlns:mat="http://www.w3.org/1998/Math/MathML" display="inline" altimg="urn:x-wiley:00246107:media:jlms12083:jlms12083-math-0001" xmlns:wiley="http://www.wiley.com/namespaces/wiley/wiley" wiley:location="equation/jlms12083-math-0001.png"> <mat:mi>D</mat:mi> </mat:math> with the following property: a set of sampling has density <mat:math xmlns:mat="http://www.w3.org/1998/Math/MathML" display="inline" altimg="urn:x-wiley:00246107:media:jlms12083:jlms12083-math-0002" xmlns:wiley="http://www.wiley.com/namespaces/wiley/wiley" wiley:location="equation/jlms12083-math-0002.png"> <mat:mrow> <mat:mo>?</mat:mo> <mat:mi>D</mat:mi> </mat:mrow> </mat:math> , whereas a set of interpolation has density <mat:math xmlns:mat="http://www.w3.org/1998/Math/MathML" display="inline" altimg="urn:x-wiley:00246107:media:jlms12083:jlms12083-math-0003" xmlns:wiley="http://www.wiley.com/namespaces/wiley/wiley" wiley:location="equation/jlms12083-math-0003.png"> <mat:mrow> <mat:mo>?</mat:mo> <mat:mi>D</mat:mi> </mat:mrow> </mat:math> . The main theorem unifies many known density theorems in signal processing, complex analysis, and harmonic analysis. For the special case of bandlimited function we recover Landa </span> <span class="z_kbtn z_kbtnclass hoverxs" style="display: none;">展开▼</span> </div> <div class="translation abstracttxt"> <span class="zhankaihshouqi fivelineshidden" id="abstract"> <span>机译：</span><Abstract XMLNS =“http://www.wiley.com/namespaces/wiley”type =“main”xml：lang =“en”> <title type =“main”>抽象</ title> >我们派生用于采样的必要密度条件和常规再现内核的内插空间，满足空间几何形状的一些自然条件和再现内核的空间。如果与球的体积相比，壳的体积小（弱环形衰减属性），并且核心具有一些非对角线衰减甚至一些较弱的定位形式，那么就存在临界密度<mat：Math XMLNS：MAT =“http://www.w3.org/1998/math/mathml”display =“内联”altimg =“urn：x-wiley：00246107：介质：jlms12083：jlms12083-math-0001”xmlns：wiley =“http ：//www.wiley.com/namespaces/wiley/wiley“wiley：location =”等式/ jlms12083-math-0001.png“> <mat：mi> d </ mat：mi> </ mat：math>与以下属性：一组采样具有密度<mat：math xmlns：mat =“http://www.w3.org/1998/math/mathml”display =“内联”Altimg =“URN：X-Wiley：00246107 ：媒体：jlms12083：jlms12083-math-0002“xmlns：wiley.com/namespaces/wiley/wiley”wiley：location =“等式/ jlms12083-math-0002.png”> <mat ：MROW> <MAT：MO>？</ MAT：MO> <MAT：MI> D </ MAT：MI> </ MAT：MROW> </ MAT：MATH>，而一组插值具有密度<MAT：数学XMLNS：MAT =“http://www.w3.org/1998/math/mathml”display =“线“Altimg =”URN：X-Wiley：00246107：媒体：jlms12083：jlms12083-math-0003“xmlns：wiley.com/namespaces/wiley/wiley”wiley：location =“等式/jlms12083-math -0003.png“> <mat：mrow> <mat：mo>？</ mat：mo> <may：mi> d </ mat：mi> </ mat：mrow> </ mat：math >。主要定理统一信号处理，复杂分析和谐波分析中的许多已知的密度定理。对于特殊情况的带限函数，我们恢复Landa </span> <span class="z_kbtn z_kbtnclass hoverxs" style="display: none;">展开▼</span> </div> </div> <div class="record"> <h2 class="all_title" id="enpatent33" >著录项</h2> <ul> <li> <span class="lefttit">来源</span> <div style="width: 86%;vertical-align: text-top;display: inline-block;"> <a href='/journal-foreign-31023/'>《The Journal of the London Mathematical Society》</a> <b style="margin: 0 2px;">|</b><span>2017年第3期</span><b style="margin: 0 2px;">|</b><span>共24页</span> </div> </li> <li> <div class="author"> <span class="lefttit">作者</span> <p id="fAuthorthree" class="threelineshidden zhankaihshouqi"> <a href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=Führ Hartmut&option=202" target="_blank" rel="nofollow">Führ Hartmut;</a> <a href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=Gr?chenig Karlheinz&option=202" target="_blank" rel="nofollow">Gr?chenig Karlheinz;</a> <a href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=Haimi Antti&option=202" target="_blank" rel="nofollow">Haimi Antti;</a> <a href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=Klotz Andreas&option=202" target="_blank" rel="nofollow">Klotz Andreas;</a> <a href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=Romero José Luis&option=202" target="_blank" rel="nofollow">Romero José Luis;</a> </p> <span class="z_kbtnclass z_kbtnclassall hoverxs" id="zkzz" style="display: none;">展开▼</span> </div> </li> <li> <div style="display: flex;"> <span class="lefttit">作者单位</span> <div style="position: relative;margin-left: 3px;max-width: 639px;"> <div class="threelineshidden zhankaihshouqi" id="fOrgthree"> <p>Lehrstuhl A für MathematikRWTH AachenD‐52056 Aachen Germany;</p> <p>Faculty of MathematicsUniversity of ViennaOskar‐Morgenstern‐Platz 1 A‐1090 Vienna Austria;</p> <p>Faculty of MathematicsUniversity of ViennaOskar‐Morgenstern‐Platz 1 A‐1090 Vienna Austria;</p> <p>Faculty of MathematicsUniversity of ViennaOskar‐Morgenstern‐Platz 1 A‐1090 Vienna Austria;</p> <p>Faculty of MathematicsUniversity of ViennaOskar‐Morgenstern‐Platz 1 A‐1090 Vienna Austria;</p> </div> <span class="z_kbtnclass z_kbtnclassall hoverxs" id="zhdw" style="display: none;">展开▼</span> </div> </div> </li> <li > <span class="lefttit">收录信息</span> <span style="width: 86%;vertical-align: text-top;display: inline-block;"></span> </li> <li> <span class="lefttit">原文格式</span> <span>PDF</span> </li> <li> <span class="lefttit">正文语种</span> <span>eng</span> </li> <li> <span class="lefttit">中图分类</span> <span><a href="https://www.zhangqiaokeyan.com/clc/156.html" title="数学">数学;</a></span> </li> <li class="antistop"> <span class="lefttit">关键词</span> <p style="width: 86%;vertical-align: text-top;"> <a style="color: #3E7FEB;" href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=42C15&option=203" rel="nofollow">42C15;</a> <a style="color: #3E7FEB;" href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=42C30 (primary)&option=203" rel="nofollow">42C30 (primary);</a> <a style="color: #3E7FEB;" href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=94A12&option=203" rel="nofollow">94A12;</a> <a style="color: #3E7FEB;" href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=46C05&option=203" rel="nofollow">46C05;</a> <a style="color: #3E7FEB;" href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=32A70 (secondary)&option=203" rel="nofollow">32A70 (secondary);</a> </p> <div class="translation"> 机译：42C15;42C30（初级）;94A12;46C05;32A70（二级）; </div> </li> </ul> </div> </div> <div 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classification by kernel density shape interpolation of clusters</a> <b>[P]</b> . <span> 外国专利：  US7542954B1 </span> <span> . 2009-06-02</span> </div> <p class="zwjiyix translation" style="max-width: initial;height: auto;word-break: break-all;white-space: initial;text-overflow: initial;overflow: initial;"> <span>机译：通过聚类的核密度形状插值进行数据分类 </span> </p> </li> <li> <div> <b>5. </b><a class="enjiyixqcontent" href="/patent-detail/06130430464230.html">Data classification by kernel density shape interpolation of clusters</a> <b>[P]</b> . <span> 外国专利：  US7542953B1 </span> <span> . 2009-06-02</span> </div> <p class="zwjiyix translation" style="max-width: initial;height: auto;word-break: break-all;white-space: initial;text-overflow: initial;overflow: initial;"> <span>机译：通过聚类的核密度形状插值进行数据分类 </span> </p> </li> </ul> </div> </div> </div> <div class="theme cardcommon" style="overflow: auto;display:none"> <h3 class="all_title" id="enpatent55">相关主题</h3> <ul id="subject"> </ul> </div> </div> </div> </div> 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