# On the speed of uniform convergence in Mercer's theorem

arXiv: https://arxiv.org/abs/2205.00487 · HTML: https://mlsanity.com/paper/2205.00487/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 1 of its 17 references that are in the map.
- First posted on arXiv 1 May 2022.

## Builds on (most cited first, 1 of 1)

- [Reproducing kernel Hilbert spaces on manifolds: Sobolev and Diffusion spaces](https://mlsanity.com/paper/1905.10913/) (2019)

## Built on by (most cited first, 5 of 5)

- [On the Intrinsic Dimensions of Data in Kernel Learning](https://mlsanity.com/paper/2601.16139/) (2026)
- [Gaussian kernel expansion with basis functions uniformly bounded in L∞](https://mlsanity.com/paper/2410.01394/) (2024)
- [Adaptive prediction theory combining offline and online learning](https://mlsanity.com/paper/2512.00342/) (2025)
- [Dynamic Regret for Online Regression in RKHS via Discounted VAW and Subspace Approximation](https://mlsanity.com/paper/2604.25021/) (2026)
- [Conditional KRR: Injecting Unpenalized Features into Kernel Methods with Applications to Kernel Thresholding](https://mlsanity.com/paper/2605.26067/) (2026)

## Abstract (arXiv)

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 sumi=1^∞ λiφi(mathbf x)φi(mathbf 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 Obig((sumi=N+1^∞λi)^fracmm+nbig) or Obig((sumi=N+1^∞λ2i)^fracm2m+nbig). Finally, we demonstrate some applications of our results to a spectral charaterization of integral operators with continuous roots and other powers.