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arXiv · 2208.14779

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

Ricardo Carrizo Vergara
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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 $D$.

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Ricardo Carrizo Vergara. 2022-08-31. Necessary and sufficient conditions for a family of continuous functions to form a Karhunen-Lo\`eve basis. https://arxiv.org/abs/2208.14779

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