On the speed of uniform convergence in Mercer's theorem

2022/05/01 by Rustem Takhanov, Takhanov, Rustem · 1 citation
Mathematics · Computer Science · #Numerical methods in inverse problems #Mathematical functions and polynomials #Matrix Theory and Algorithms

paper · pdf · doi:10.48550/arxiv.2205.00487

Abstract

The classical Mercer's theorem claims that a continuous positive definite kernel K(\mathbf x, \mathbf y) on a compact set can be represented as ∑_i=1^∞ λ_iϕ_i(\mathbf x)ϕ_i(\mathbf y) where \(λ_i,ϕ_i)\ 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 O((∑_i=N+1^∞λ_i)^(m)/(m+n)) or O((∑_i=N+1^∞λ²_i)^(m)/(2m+n)). Finally, we demonstrate some applications of our results to a spectral charaterization of integral operators with continuous roots and other powers.

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