Necessary and sufficient conditions for a family of continuous functions to form a Karhunen-Lo\`eve 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 \leq n} \lambda_{j} f_{j}^{2} $ is equicontinuous over $D$.