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Autumn 2016 Positive definite kernels and boundary spaces
Palle Jorgensen, Feng Tian
Adv. Oper. Theory 1(1): 123-133 (Autumn 2016). DOI: 10.22034/aot.1610.1044

Abstract

We consider a kernel based harmonic analysis of "boundary," and boundary representations. Our setting is general: certain classes of positive definite kernels. Our theorems extend (and are motivatedby) results and notions from classical harmonic analysis on the disk. Our positive definite kernels include those defined on infinite discrete sets, for example sets of vertices in electrical networks, or discrete sets which arise from sampling operations performed on positive definite kernels in a continuous setting.

Below we give a summary of main conclusions in the paper: Starting with a given positive definite kernel $K$ we make precise generalized boundaries for $K$. They are measure theoretic "boundaries." Using the theory of Gaussian processes, we show that there is always such a generalized boundary for any positive definite kernel.

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Palle Jorgensen. Feng Tian. "Positive definite kernels and boundary spaces." Adv. Oper. Theory 1 (1) 123 - 133, Autumn 2016. https://doi.org/10.22034/aot.1610.1044

Information

Received: 29 October 2016; Accepted: 29 November 2016; Published: Autumn 2016
First available in Project Euclid: 4 December 2017

zbMATH: 1359.42004
MathSciNet: MR3721329
Digital Object Identifier: 10.22034/aot.1610.1044

Subjects:
Primary: 47L60
Secondary: 22E70 , 46N20

Keywords: discrete analysis , Gaussian free fields , Green’s function , non-uniform sampling , ‎reproducing kernel Hilbert ‎space

Rights: Copyright © 2016 Tusi Mathematical Research Group

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