On the speed of uniform convergence in Mercer's theorem

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 $$\sum_{i=1}^\infty \lambda_i\phi_i({\mathbf x})\phi_i({\mathbf y})$$ where ${(\lambda_i,\phi_i)}$ 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 $$\mathcal{O}\big((\sum_{i=N+1}<sup>\infty\lambda_i)<sup>{\frac{m}{m+n}}\big)$$ or $$\mathcal{O}\big((\sum_{i=N+1}<sup>\infty\lambda<sup>2_i)<sup>{\frac{m}{2m+n}}\big)$$. Finally, we demonstrate some applications of our results to a spectral charaterization of integral operators with continuous roots and other powers.