On the speed of uniform convergence in Mercer's theorem

Mercer kernels play an important role in machine learning and is a mathematical basis of such techniques as kernel density estimation and spline models [14], Support Vector Machines [11], kernel principal components analysis [10], regularization of neural networks [13] and many others. According to Aronszajn's theorem, any Mercer kernel induces a reproducing kernel Hilbert space (RKHS) and vice versa, any RKHS corresponds to a kernel. A relationship between the latter two notions is decribed in the classical Mercer's theorem. A goal of this note is torefine this theoremandgive some estimates onthe speedof uniformconvergencestated in it.