On the speed of uniform convergence in Mercer's theorem

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 $\sum{i=1}^\infty \lambdai\phii({\mathbf x})\phii({\mathbf y})$ where ${(\lambdai,\phii)}$ 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 $\mathcal{O}\big((\sum{i=N+1}^{\infty\lambda}i)^{{\frac{m}{m+n}}\big)$} or $\mathcal{O}\big((\sum{i=N+1}^{\infty\lambda2}i)^{{\frac{m}{2m+n}}\big)$.} Finally, we demonstrate some applications of our results to a spectral charaterization of integral operators with continuous roots and other powers.