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Schoenberg Matrices of Radial Positive Definite Functions and Riesz Sequences of Translates in \(L^2({\mathbb R}^n)\)

Abstract

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 \(g\in L^2({\mathbb R}^n)\).

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Acknowledgments

We are grateful to T. Gneiting for comments on \(\alpha \)-stable functions and the Matérn classes, A. Kheifets for the function theoretic argument in the proof of Lemma 3.23, B. Mityagin and V. Zastavnyi for careful reading of the manuscript and valuable remarks.

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Correspondence to L. Golinskii.

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Communicated by David Walnut.

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Golinskii, L., Malamud, M. & Oridoroga, L. Schoenberg Matrices of Radial Positive Definite Functions and Riesz Sequences of Translates in \(L^2({\mathbb R}^n)\) . J Fourier Anal Appl 21, 915–960 (2015). https://doi.org/10.1007/s00041-015-9391-4

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  • DOI: https://doi.org/10.1007/s00041-015-9391-4

Keywords

  • Infinite matrices
  • Schur test
  • Riesz sequences
  • Completely monotone functions
  • Fourier transform
  • Gram matrices
  • Minimal sequences
  • Toeplitz operators

Mathematics Subject Classification

  • 42A82
  • 42B10
  • 33C10
  • 47B37