## 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=1}^{∞}λ_{i}ϕ_{i}(x)ϕ_{i}(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 [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 language | English |
---|---|

Article number | 126718 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 518 |

Issue number | 2 |

DOIs | |

Publication status | Published - Feb 15 2023 |

## Keywords

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

## ASJC Scopus subject areas

- Analysis
- Applied Mathematics