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 $D$.
Categories: math.FA math.PR