Equivalence of modes of convergence on reproducing kernel Hilbert spaces

TL;DR

This paper proves that pointwise, weak, and strong convergence modes in RKHS are equivalent, leading to insights on positive operators, kernels, and embedding conditions.

Contribution

It establishes the equivalence of various convergence modes in RKHS and explores their implications for operators and embeddings.

Findings

Modes of convergence in RKHS are equivalent.

Established connection between convergence and positive operators.

Derived compact embedding conditions for RKHS in L2 spaces.

Abstract

Let $(X,\mu)$ be a strictly-positive Borel measure space. We show that the modes of convergence in a reproducing kernel Hilbert (RKHS) space, pointwise, weak and strong are all equivalents. From this we describe some important consequences such as an association with positive operators and positive definite kernels and a compact embedding condition for a RKHS in $L^{2}(X,\mu)$.