Necessary and sufficient conditions for a family of continuous functions to form a Karhunen-Loève basis

Abstract

Given an orthonormal system of $L^2(D)$ consistent of continuous functions $(f_n{)}_n$, with $D\subset {\mathbb{R}}^d$ compact, and given a sequence of strictly positive coefficients $({\lambda }_n{)}_n$ forming a convergent series, we prove that they consist in the eigenfunctions and eigenvectors of a covariance operator associated to a continuous positive-definite Kernel if and only if the sequence of partial sums ${\sum }_{j\le n}{\lambda }_j{f}_j^2$ is equicontinuous over $D$.

3 pages short result