{"title":"用多项式逼近两个矩阵的解析函数的估计值","authors":"V. G. Kurbatov, I. V. Kurbatova","doi":"10.1134/s1995080224601206","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>Let <span>\\(U,V\\subseteq\\mathbb{C}\\)</span> be open convex sets, and <span>\\(z_{1}\\)</span>,\n<span>\\(z_{2}\\)</span>, <span>\\(\\dots,z_{N}\\in U\\)</span> and <span>\\(w_{1}\\)</span>, <span>\\(w_{2}\\)</span>, <span>\\(\\dots,w_{M}\\in V\\)</span> be\n(maybe repetitive) points. Let <span>\\(f:\\,U\\times V\\to\\mathbb{C}\\)</span> be an\nanalytic function. Let the interpolating polynomial <span>\\(p\\)</span> be\ndetermined by the values of <span>\\(f\\)</span> on the rectangular grid\n<span>\\((z_{i},w_{j})\\)</span>, <span>\\(i=1,2,\\dots,N\\)</span>, <span>\\(j=1,2,\\dots,M\\)</span>. Let <span>\\(A\\)</span> and <span>\\(B\\)</span>\nbe matrices of the sizes <span>\\(n\\times n\\)</span> and <span>\\(m\\times m\\)</span>,\nrespectively. The function <span>\\(f\\)</span> of <span>\\(A\\)</span> and <span>\\(B\\)</span> can be defined by\nthe formula</p><span>$$f(A,B)=\\frac{1}{(2\\pi i)^{2}}\\int\\limits_{\\Gamma_{1}}\\int\\limits_{\\Gamma_{2}}f(\\lambda,\\mu)(\\lambda\\mathbf{1}-A)^{-1}\\otimes(\\mu\\mathbf{1}-B)^{-1}\\,d\\mu\\,d\\lambda,$$</span><p>where <span>\\(\\Gamma_{1}\\)</span> and <span>\\(\\Gamma_{2}\\)</span> surround the spectra <span>\\(\\sigma(A)\\)</span>\nand <span>\\(\\sigma(B)\\)</span>, respectively; <span>\\(p(A,B)\\)</span> is defined in the same\nway. An estimate of <span>\\(||f(A,B)-p(A,B)||\\)</span> is given.</p>","PeriodicalId":46135,"journal":{"name":"Lobachevskii Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8000,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"An Estimate of Approximation of an Analytic Function of Two Matrices by a Polynomial\",\"authors\":\"V. G. Kurbatov, I. V. Kurbatova\",\"doi\":\"10.1134/s1995080224601206\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<h3 data-test=\\\"abstract-sub-heading\\\">Abstract</h3><p>Let <span>\\\\(U,V\\\\subseteq\\\\mathbb{C}\\\\)</span> be open convex sets, and <span>\\\\(z_{1}\\\\)</span>,\\n<span>\\\\(z_{2}\\\\)</span>, <span>\\\\(\\\\dots,z_{N}\\\\in U\\\\)</span> and <span>\\\\(w_{1}\\\\)</span>, <span>\\\\(w_{2}\\\\)</span>, <span>\\\\(\\\\dots,w_{M}\\\\in V\\\\)</span> be\\n(maybe repetitive) points. Let <span>\\\\(f:\\\\,U\\\\times V\\\\to\\\\mathbb{C}\\\\)</span> be an\\nanalytic function. Let the interpolating polynomial <span>\\\\(p\\\\)</span> be\\ndetermined by the values of <span>\\\\(f\\\\)</span> on the rectangular grid\\n<span>\\\\((z_{i},w_{j})\\\\)</span>, <span>\\\\(i=1,2,\\\\dots,N\\\\)</span>, <span>\\\\(j=1,2,\\\\dots,M\\\\)</span>. Let <span>\\\\(A\\\\)</span> and <span>\\\\(B\\\\)</span>\\nbe matrices of the sizes <span>\\\\(n\\\\times n\\\\)</span> and <span>\\\\(m\\\\times m\\\\)</span>,\\nrespectively. The function <span>\\\\(f\\\\)</span> of <span>\\\\(A\\\\)</span> and <span>\\\\(B\\\\)</span> can be defined by\\nthe formula</p><span>$$f(A,B)=\\\\frac{1}{(2\\\\pi i)^{2}}\\\\int\\\\limits_{\\\\Gamma_{1}}\\\\int\\\\limits_{\\\\Gamma_{2}}f(\\\\lambda,\\\\mu)(\\\\lambda\\\\mathbf{1}-A)^{-1}\\\\otimes(\\\\mu\\\\mathbf{1}-B)^{-1}\\\\,d\\\\mu\\\\,d\\\\lambda,$$</span><p>where <span>\\\\(\\\\Gamma_{1}\\\\)</span> and <span>\\\\(\\\\Gamma_{2}\\\\)</span> surround the spectra <span>\\\\(\\\\sigma(A)\\\\)</span>\\nand <span>\\\\(\\\\sigma(B)\\\\)</span>, respectively; <span>\\\\(p(A,B)\\\\)</span> is defined in the same\\nway. An estimate of <span>\\\\(||f(A,B)-p(A,B)||\\\\)</span> is given.</p>\",\"PeriodicalId\":46135,\"journal\":{\"name\":\"Lobachevskii Journal of Mathematics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.8000,\"publicationDate\":\"2024-08-22\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Lobachevskii Journal of Mathematics\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1134/s1995080224601206\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Lobachevskii Journal of Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1134/s1995080224601206","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
AbstractLet\(U,Vsubseteq\mathbb{C}\) be open convex sets, and \(z_{1}\),\(z_{2}\),\(\dots,z_{N}\in U\) and \(w_{1}\),\(w_{2}\),\(\dots,w_{M}\in V\) be (maybe repetitive) points.让(f:\,U\times Vto\mathbb{C}\) 是一个解析函数。让内插多项式(p)由矩形网格((z_{i},w_{j}))上的(f)值决定,(i=1,2,(dots,N),(j=1,2,(dots,M))。让(A)和(B)分别是大小为(n乘以n)和(m乘以m)的矩阵。公式$f(A,B)=frac{1}{(2\pi i)^{2}}\intlimits_{\Gamma_{1}}\intlimits_{\Gamma_{2}}f(\lambda、\其中 \(\Gamma_{1}\) 和 \(\Gamma_{2}\) 分別圍繞著 \(\sigma(A)\) 和 \(\sigma(B)\) 的光譜;\p(A,B)的定义是一样的。给出了 \(||f(A,B)-p(A,B)||\) 的估计值。
An Estimate of Approximation of an Analytic Function of Two Matrices by a Polynomial
Abstract
Let \(U,V\subseteq\mathbb{C}\) be open convex sets, and \(z_{1}\),
\(z_{2}\), \(\dots,z_{N}\in U\) and \(w_{1}\), \(w_{2}\), \(\dots,w_{M}\in V\) be
(maybe repetitive) points. Let \(f:\,U\times V\to\mathbb{C}\) be an
analytic function. Let the interpolating polynomial \(p\) be
determined by the values of \(f\) on the rectangular grid
\((z_{i},w_{j})\), \(i=1,2,\dots,N\), \(j=1,2,\dots,M\). Let \(A\) and \(B\)
be matrices of the sizes \(n\times n\) and \(m\times m\),
respectively. The function \(f\) of \(A\) and \(B\) can be defined by
the formula
where \(\Gamma_{1}\) and \(\Gamma_{2}\) surround the spectra \(\sigma(A)\)
and \(\sigma(B)\), respectively; \(p(A,B)\) is defined in the same
way. An estimate of \(||f(A,B)-p(A,B)||\) is given.
期刊介绍:
Lobachevskii Journal of Mathematics is an international peer reviewed journal published in collaboration with the Russian Academy of Sciences and Kazan Federal University. The journal covers mathematical topics associated with the name of famous Russian mathematician Nikolai Lobachevsky (Lobachevskii). The journal publishes research articles on geometry and topology, algebra, complex analysis, functional analysis, differential equations and mathematical physics, probability theory and stochastic processes, computational mathematics, mathematical modeling, numerical methods and program complexes, computer science, optimal control, and theory of algorithms as well as applied mathematics. The journal welcomes manuscripts from all countries in the English language.