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Square summability with geometric weight for classical orthogonal expansions

D. Karp
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Abstract

Let $f_k$ be the $k$-th Fourier coefficient of a function $f$ in terms of the orthonormal Hermite, Laguerre or Jacobi polynomials. We give necessary and sufficient conditions on $f$ for the inequality $\sum_{k}|f_k|^2θ^k<\infty$ to hold with $θ>1$. As a by-product new orthogonality relations for the Hermite and Laguerre polynomials are found. The basic machinery for the proofs is provided by the theory of reproducing kernel Hilbert spaces.

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D. Karp. 2006-04-03. Square summability with geometric weight for classical orthogonal expansions. https://arxiv.org/abs/math/0604028

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