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Radial positive definite functions and Schoenberg matrices with negative eigenvalues

Authors: L. Golinskii, M. Malamud and L. Oridoroga
Journal: Trans. Amer. Math. Soc. 370 (2018), 1-25
MSC (2010): Primary 42A82, 42B10, 47B37
Published electronically: May 1, 2017
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Abstract: The main object under consideration is a class $ \Phi _n\backslash \Phi _{n+1}$ of radial positive definite functions on $ \mathbb{R}^n$ which do not admit radial positive definite continuation on $ \mathbb{R}^{n+1}$. We find certain necessary and sufficient conditions on the Schoenberg representation measure $ \nu _n$ of $ f\in \Phi _n$ for $ f\in \Phi _{n+k}$, $ k\in \mathbb{N}$. We show that the class $ \Phi _n\backslash \Phi _{n+k}$ is rich enough by giving a number of examples. In particular, we give a direct proof of $ \Omega _n\in \Phi _n\backslash \Phi _{n+1}$, which avoids Schoenberg's theorem; $ \Omega _n$ is the Schoenberg kernel. We show that $ \Omega _n(a\cdot )\Omega _n(b\cdot )\in \Phi _n\backslash \Phi _{n+1}$ for $ a\not =b$. Moreover, for the square of this function we prove the surprisingly much stronger result $ \Omega _n^2(a\cdot )\in \Phi _{2n-1}\backslash \Phi _{2n}$. We also show that any $ f\in \Phi _n\backslash \Phi _{n+1}$, $ n\ge 2$, has infinitely many negative squares. The latter means that for an arbitrary positive integer $ N$ there is a finite Schoenberg matrix $ \mathcal {S}_X(f) := \Vert f(\vert x_i-x_j\vert _{n+1})\Vert _{i,j=1}^{m}$, $ X := \{x_j\}_{j=1}^m \subset \mathbb{R}^{n+1}$, which has at least $ N$ negative eigenvalues.

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Additional Information

L. Golinskii
Affiliation: B. Verkin Institute for Low Temperature Physics and Engineering, 47 Science Avenue, 61103 Kharkiv, Ukraine

M. Malamud
Affiliation: Institute of Applied Mathematics and Mechanics, NAS of Ukraine, Batyuka Street, 19, Slavyansk, Ukraine – and – People’s Friendship University of Russia, Miklukho-Maklaya Street 6, Moscow 117198, Russia

L. Oridoroga
Affiliation: Donetsk National University, 24, Universitetskaya Street, 83055 Donetsk, Ukraine

Keywords: Positive definite functions, Schoenberg kernels, hypergeometric functions, Hankel transform, Bessel functions
Received by editor(s): April 7, 2015
Received by editor(s) in revised form: October 11, 2015
Published electronically: May 1, 2017
Additional Notes: The second author’s research was supported by the Ministry of Education and Science of the Russian Federation (Agreement No. 02.a03.21.0008)
Article copyright: © Copyright 2017 American Mathematical Society

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