Mercer's expansion and Mercer's theorem are cornerstone results in kernel theory. While the classical Mercer's theorem only considers continuous symmetric positive definite kernels, analogous expansions are effective in practice for indefinite and asymmetric kernels. In this paper we extend Mercer's expansion to continuous kernels, providing a rigorous theoretical underpinning for indefinite and asymmetric kernels. We begin by demonstrating that Mercer's expansion may not be pointwise convergent for continuous indefinite kernels, before proving that the expansion of continuous kernels with bounded variation uniformly in each variable separably converges pointwise almost everywhere, almost uniformly, and unconditionally almost everywhere. We also describe an algorithm for computing Mercer's expansion for general kernels and give new decay bounds on its terms.

翻译：Mercer展开与Mercer定理是核理论中的基石性成果。经典Mercer定理仅针对连续对称正定核，然而在实际应用中，类似展开对不定核与非对称核同样有效。本文提出将Mercer展开推广至连续核，为不定核与非对称核建立严格的理论基础。我们首先证明连续不定核的Mercer展开可能不满足逐点收敛性，随后论证各变量分别具有有界变差的连续核展开式在几乎处处意义下逐点收敛、几乎一致收敛，且在几乎处处意义下无条件收敛。本文还提出一种适用于一般核的Mercer展开计算算法，并给出其展开项衰减速率的新界。