Schoenberg Matrices of Radial Positive Definite Functions and Riesz Sequences of Translates in 详细信息    查看全文

• 作者：L. Golinskii ; M. Malamud ; L. Oridoroga
• 关键词：Infinite matrices ; Schur test ; Riesz sequences ; Completely monotone functions ; Fourier transform ; Gram matrices ; Minimal sequences ; Toeplitz operators ; 42A82 ; 42B10 ; 33C10 ; 47B37
• 刊名：Journal of Fourier Analysis and Applications
• 出版年：2015
• 出版时间：October 2015
• 年：2015
• 卷：21
• 期：5
• 页码：915-960
• 全文大小：1,149 KB
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• 作者单位：L. Golinskii (1)
  M. Malamud (2)
  L. Oridoroga (3)
  
  1. Mathematics Division, Low Temperature Physics Institute, NAS of Ukraine, 47 Lenin Ave., Kharkov, 61103, Ukraine
  2. Institute of Applied Mathematics and Mechanics, NAS of Ukraine, 74 R. Luxemburg Str., Donetsk, 83114, Ukraine
  3. Donetsk National University, 24, Universitetskaya Str., Donetsk, 83055, Ukraine
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Fourier Analysis
  Abstract Harmonic Analysis
  Approximations and Expansions
  Partial Differential Equations
  Applications of Mathematics
  Signal,Image and Speech Processing
  
• 出版者：Birkh盲user Boston
• ISSN：1531-5851

文摘

Given a function \(f\) on the positive half-line \({\mathbb R}_+\) and a sequence (finite or infinite) of points \(X=\{x_k\}_{k=1}^\omega \) in \({\mathbb R}^n\), we define and study matrices \({\mathcal S}_X(f)=[f(\Vert x_i-x_j\Vert )]_{i,j=1}^\omega \) called Schoenberg’s matrices. We are primarily interested in those matrices which generate bounded and invertible linear operators \(S_X(f)\) on \(\ell ^2({\mathbb N})\). We provide conditions on \(X\) and \(f\) for the latter to hold. If \(f\) is an \(\ell ^2\)-positive definite function, such conditions are given in terms of the Schoenberg measure \(\sigma _f\). Examples of Schoenberg’s operators with various spectral properties are presented. We also approach Schoenberg’s matrices from the viewpoint of harmonic analysis on \({\mathbb R}^n\), wherein the notion of the strong \(X\)-positive definiteness plays a key role. In particular, we prove that each radial \(\ell ^2\) -positive definite function is strongly \(X\) -positive definite whenever \(X\) is a separated set. We also implement a “grammization-procedure for certain positive definite Schoenberg’s matrices. This leads to Riesz–Fischer and Riesz sequences (Riesz bases in their linear span) of the form \({\mathcal F}_X(g)=\{g(\cdot -x_j)\}_{x_j\in X}\) for certain radial functions \(g\in L^2({\mathbb R}^n)\). Keywords Infinite matrices Schur test Riesz sequences Completely monotone functions Fourier transform Gram matrices Minimal sequences Toeplitz operators