On the speed of uniform convergence in Mercer's theorem

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2 Citations (Scopus)

Abstract

The classical Mercer's theorem claims that a continuous positive definite kernel K(x,y) on a compact set can be represented as ∑i=1λiϕi(x)ϕi(y) where {(λii)} are eigenvalue-eigenvector pairs of the corresponding integral operator. This infinite representation is known to converge uniformly to the kernel K. We estimate the speed of this convergence in terms of the decay rate of eigenvalues and demonstrate that for 2m times differentiable kernels the first N terms of the series approximate K as [Formula presented] or [Formula presented]. Finally, we demonstrate some applications of our results to a spectral characterization of integral operators with continuous roots and other powers.

Original languageEnglish
Article number126718
JournalJournal of Mathematical Analysis and Applications
Volume518
Issue number2
DOIs
Publication statusPublished - Feb 15 2023

Keywords

  • Gagliardo-Nirenberg inequality
  • Mercer kernel
  • Mercer's theorem
  • RKHS
  • Uniform convergence

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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