Necessary and sufficient conditions for a family of continuous functions to form a Karhunen-Lo\`eve basis

TL;DR

This paper establishes necessary and sufficient conditions for a family of continuous functions to form a Karhunen-Loève basis, linking eigenfunctions of a covariance operator to equicontinuity of partial sums.

Contribution

It provides a precise characterization of when an orthonormal system of continuous functions forms a Karhunen-Loève basis based on equicontinuity conditions.

Findings

Characterization of Karhunen-Loève bases via equicontinuity.

Necessary and sufficient conditions for eigenfunction representation.

Link between covariance operators and continuous function systems.

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 \leq n} \lambda_{j} f_{j}^{2} $ is equicontinuous over $D$.