On the speed of uniform convergence in Mercer's theorem

• URL: http://arxiv.org/abs/2205.00487v1
• Date: Sun, 1 May 2022 15:07:57 GMT
• Title: On the speed of uniform convergence in Mercer's theorem
• Authors: Rustem Takhanov
• Abstract summary: A continuous positive definite kernel $K(mathbf x, mathbf y)$ on a compact set can be represented as $sum_i=1infty lambda_iphi_i(mathbf x)phi_i(mathbf y)$ where $(lambda_i,phi_i)$ are eigenvalue-eigenvector pairs of the corresponding integral operator. We estimate the speed of this convergence in terms of the decay rate of eigenvalues and demonstrate that for $3m
• Score: 6.028247638616059
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 $3m$ 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 $\mathcal{O}\big((\sum_{i=N+1}^\infty\lambda^2_i)^{\frac{m}{2m+n}}\big)$.

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