On the speed of uniform convergence in Mercers theorem
Takhanov R.
15 February 2023Academic Press Inc.
Journal of Mathematical Analysis and Applications
2023#518Issue 2
The classical Mercers theorem claims that a continuous positive definite kernel K(x,y) on a compact set can be represented as ∑i=1∞λiϕi(x)ϕi(y) where {(λi,ϕi)} 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.
Gagliardo-Nirenberg inequality , Mercer kernel , Mercers theorem , RKHS , Uniform convergence
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