On multivariate discrete least squares

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• 作者：Yeon Ju Lee^a ; ^{leeyeonju08@korea.ac.kr" class="auth_mail" title="E-mail the corresponding author} ; Charles A. Micchelli^b ; Jungho Yoon^c
• 关键词：Collocation matrix ; Wronskian ; Multivariate least squares ; Multivariate Maclaurin expansion
• 刊名：Journal of Approximation Theory
• 出版年：2016
• 出版时间：November 2016
• 年：2016
• 卷：211
• 期：Complete
• 页码：78-84
• 全文大小：207 K

文摘

For a positive integer class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021904516300521&_mathId=si1.gif&_user=111111111&_pii=S0021904516300521&_rdoc=1&_issn=00219045&md5=b75894cb2fa6e6733ab90fb853064253" title="Click to view the MathML source">n∈Nclass="mathContainer hidden">class="mathCode">th altimg="si1.gif" overflow="scroll">n∈thvariant="double-struck">Nth> we introduce the index set class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021904516300521&_mathId=si2.gif&_user=111111111&_pii=S0021904516300521&_rdoc=1&_issn=00219045&md5=55667a62c31f2135084ec84a378f8130" title="Click to view the MathML source">N_n:={1,2,…,n}class="mathContainer hidden">class="mathCode">th altimg="si2.gif" overflow="scroll">thvariant="double-struck">Nn:={1,2,…,n}th>. Let class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021904516300521&_mathId=si3.gif&_user=111111111&_pii=S0021904516300521&_rdoc=1&_issn=00219045&md5=3f1942b80f52411a4fa63a7de1e69a98" title="Click to view the MathML source">X:={x_i:i∈N_n}class="mathContainer hidden">class="mathCode">th altimg="si3.gif" overflow="scroll">X:={xi:i∈thvariant="double-struck">Nn}th> be a distinct   set of vectors in class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021904516300521&_mathId=si4.gif&_user=111111111&_pii=S0021904516300521&_rdoc=1&_issn=00219045&md5=b834d8b867d6dc8d253b603d8fefdb06" title="Click to view the MathML source">R^dclass="mathContainer hidden">class="mathCode">th altimg="si4.gif" overflow="scroll">thvariant="double-struck">Rdth>, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021904516300521&_mathId=si5.gif&_user=111111111&_pii=S0021904516300521&_rdoc=1&_issn=00219045&md5=0afb3835fd52f12e85daedaf056622a9" title="Click to view the MathML source">Y:={y_i:i∈N_n}class="mathContainer hidden">class="mathCode">th altimg="si5.gif" overflow="scroll">Y:={yi:i∈thvariant="double-struck">Nn}th> a prescribed   data set of real numbers in class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021904516300521&_mathId=si6.gif&_user=111111111&_pii=S0021904516300521&_rdoc=1&_issn=00219045&md5=7b78d460a1edf968cbfd82605d137085" title="Click to view the MathML source">Rclass="mathContainer hidden">class="mathCode">th altimg="si6.gif" overflow="scroll">thvariant="double-struck">Rth> and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021904516300521&_mathId=si7.gif&_user=111111111&_pii=S0021904516300521&_rdoc=1&_issn=00219045&md5=7c9862baab9a428eb13ce892a0a77028" title="Click to view the MathML source">F:={f_j:j∈N_m},m<nclass="mathContainer hidden">class="mathCode">th altimg="si7.gif" overflow="scroll">thvariant="script">F:={fj:j∈thvariant="double-struck">Nm},m<nth>, a given set of real valued continuous   functions defined on some neighborhood class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021904516300521&_mathId=si8.gif&_user=111111111&_pii=S0021904516300521&_rdoc=1&_issn=00219045&md5=2cd2aaf4794b02a5c1dfaec5c49e87e2" title="Click to view the MathML source">Oclass="mathContainer hidden">class="mathCode">th altimg="si8.gif" overflow="scroll">thvariant="script">Oth> of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021904516300521&_mathId=si4.gif&_user=111111111&_pii=S0021904516300521&_rdoc=1&_issn=00219045&md5=b834d8b867d6dc8d253b603d8fefdb06" title="Click to view the MathML source">R^dclass="mathContainer hidden">class="mathCode">th altimg="si4.gif" overflow="scroll">thvariant="double-struck">Rdth> containing class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021904516300521&_mathId=si10.gif&_user=111111111&_pii=S0021904516300521&_rdoc=1&_issn=00219045&md5=acb07db3bf09ad469c6a2277ac96c14a" title="Click to view the MathML source">Xclass="mathContainer hidden">class="mathCode">th altimg="si10.gif" overflow="scroll">Xth>. The discrete least squares problem determines a (generally unique) function class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021904516300521&_mathId=si11.gif&_user=111111111&_pii=S0021904516300521&_rdoc=1&_issn=00219045&md5=5be1ffe1abad5a8584bfb250a8dad9aa">class="imgLazyJSB inlineImage" height="17" width="152" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0021904516300521-si11.gif">cript>t="17" border="0" style="vertical-align:bottom" width="152" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0021904516300521-si11.gif">cript>class="mathContainer hidden">class="mathCode">th altimg="si11.gif" overflow="scroll">f=∑j∈thvariant="double-struck">Nmcj⋆fj∈text>spantext>thvariant="script">Fth> which minimizes the square of the class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021904516300521&_mathId=si12.gif&_user=111111111&_pii=S0021904516300521&_rdoc=1&_issn=00219045&md5=3ee03781ab23930bf7fbefa88e5b11f8" title="Click to view the MathML source">ℓ²−class="mathContainer hidden">class="mathCode">th altimg="si12.gif" overflow="scroll">ℓ2−th>norm over all vectors class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021904516300521&_mathId=si14.gif&_user=111111111&_pii=S0021904516300521&_rdoc=1&_issn=00219045&md5=0d2ad6f2b4ea6db167611bcdacf4523b" title="Click to view the MathML source">(c_j:j∈N_m)∈R^mclass="mathContainer hidden">class="mathCode">th altimg="si14.gif" overflow="scroll">(cj:j∈thvariant="double-struck">Nm)∈thvariant="double-struck">Rmth>. The value of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021904516300521&_mathId=si15.gif&_user=111111111&_pii=S0021904516300521&_rdoc=1&_issn=00219045&md5=839f2557f0501040d5e451a5d4b2ef4d" title="Click to view the MathML source">fclass="mathContainer hidden">class="mathCode">th altimg="si15.gif" overflow="scroll">fth> at some class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021904516300521&_mathId=si16.gif&_user=111111111&_pii=S0021904516300521&_rdoc=1&_issn=00219045&md5=07bfbc62c3ed5dce0dabb14a977c5a7a" title="Click to view the MathML source">s∈Oclass="mathContainer hidden">class="mathCode">th altimg="si16.gif" overflow="scroll">s∈thvariant="script">Oth> may be viewed as the optimally predicted value (in the class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021904516300521&_mathId=si12.gif&_user=111111111&_pii=S0021904516300521&_rdoc=1&_issn=00219045&md5=3ee03781ab23930bf7fbefa88e5b11f8" title="Click to view the MathML source">ℓ²−class="mathContainer hidden">class="mathCode">th altimg="si12.gif" overflow="scroll">ℓ2−th>sense) of all   functions in class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021904516300521&_mathId=si18.gif&_user=111111111&_pii=S0021904516300521&_rdoc=1&_issn=00219045&md5=42cac639371ec6e0b574ea9b3ef9aae5" title="Click to view the MathML source">spanFclass="mathContainer hidden">class="mathCode">th altimg="si18.gif" overflow="scroll">text>spantext>thvariant="script">Fth> from the given data class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021904516300521&_mathId=si19.gif&_user=111111111&_pii=S0021904516300521&_rdoc=1&_issn=00219045&md5=7bf858ae87b288661660a3ee7a630926" title="Click to view the MathML source">X={x_i:i∈N_n}class="mathContainer hidden">class="mathCode">th altimg="si19.gif" overflow="scroll">X={xi:i∈thvariant="double-struck">Nn}th> and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021904516300521&_mathId=si20.gif&_user=111111111&_pii=S0021904516300521&_rdoc=1&_issn=00219045&md5=aec63aee94eec18b6e876599ceefa30b" title="Click to view the MathML source">Y={y_i:i∈N_n}class="mathContainer hidden">class="mathCode">th altimg="si20.gif" overflow="scroll">Y={yi:i∈thvariant="double-struck">Nn}th>.

We ask “What happens if the components of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021904516300521&_mathId=si10.gif&_user=111111111&_pii=S0021904516300521&_rdoc=1&_issn=00219045&md5=acb07db3bf09ad469c6a2277ac96c14a" title="Click to view the MathML source">Xclass="mathContainer hidden">class="mathCode">th altimg="si10.gif" overflow="scroll">Xth> and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021904516300521&_mathId=si22.gif&_user=111111111&_pii=S0021904516300521&_rdoc=1&_issn=00219045&md5=015a6ddaddf3ca234dd160ee517514ee" title="Click to view the MathML source">sclass="mathContainer hidden">class="mathCode">th altimg="si22.gif" overflow="scroll">sth> are nearly the same”. For example, when all these vectors are near the origin in class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021904516300521&_mathId=si4.gif&_user=111111111&_pii=S0021904516300521&_rdoc=1&_issn=00219045&md5=b834d8b867d6dc8d253b603d8fefdb06" title="Click to view the MathML source">R^dclass="mathContainer hidden">class="mathCode">th altimg="si4.gif" overflow="scroll">thvariant="double-struck">Rdth>. From a practical point of view this problem comes up in image analysis when we wish to obtain a new pixel value from nearby available pixel values as was done in [2], for a specified set of functions class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021904516300521&_mathId=si24.gif&_user=111111111&_pii=S0021904516300521&_rdoc=1&_issn=00219045&md5=67f5a8129137d7cb47d282e9890e849f" title="Click to view the MathML source">Fclass="mathContainer hidden">class="mathCode">th altimg="si24.gif" overflow="scroll">thvariant="script">Fth>.

This problem was satisfactorily solved in the univariate case in Section 6 of Lee and Micchelli (2013). Here, we treat the significantly more difficult multivariate case using an approach recently provided in Yeon Ju Lee, Charles A. Micchelli and Jungho Yoon (2015).