Mercer’s Theorem on General Domains: on the Interaction between Measures, Kernels, and RKHSs

Given a compact metric space X and a strictly positive Borel measure ν on X , Mercer’s classical theorem states that the spectral decomposition of a positive self-adjoint integral operator Tk : L2(ν) → L2(ν) of a continuous k yields a series representation of k in terms of the eigenvalues and -functions of Tk. An immediate consequence of this representation is that k is a (reproducing) kernel and that its reproducing kernel Hilbert space can also be described by these eigenfunctions and -functions. It is well-known that Mercer’s theorem has found important applications in various branches of mathematics including probability theory and statistics. In particular for some applications in the latter areas, however, it would be highly convenient to have a form of Mercer’s theorem for more general spaces X and kernels k. Unfortunately, all extensions of Mercer’s theorem in this direction either stick too closely to the original topological structure of X and k, or replace the absolute and uniform convergence by weaker notions of convergence that are not strong enough for many statistical applications. In this work we fill this gap by establishing several Mercer type series representations for k that, on the one hand, make only very mild assumptions on X and k, and, on the other hand, provide convergence results that are strong enough for interesting applications in e.g. statistical learning theory. To illustrate the latter, we first use these series representations to describe ranges of fractional powers of Tk in terms of interpolations spaces and investigate under which conditions these interpolation spaces are contained in L∞(ν). For these two results we then discuss applications related to the analysis of so-called least squares support vector machines, which are a state-of-the-art learning algorithm. Besides these results we further use the obtained Mercer representations to show that every self-adjoint nuclear operator L2(ν)→ L2(ν) is an integral operator whose representing function k is the difference of two (reproducing) kernels.