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Hardy-type theorem for functions orthogonal with respect to their zeros. The Jacobi weight case

L. D. Abreu·F. Marcellan·S. Yakubovich
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Abstract

Motivated by G. H. Hardy's 1939 results \cite{Hardy} on functions orthogonal with respect to their real zeros $λ_{n}, n=1,2,... $, we will consider, within the same general conditions imposed by Hardy, functions satisfying an orthogonality with respect to their zeros with Jacobi weights on the interval $(0,1)$, that is, the functions $f(z)=z^νF(z), ν\in \mathbb{R}$, where $F$ is entire and \begin{equation*} \int_{0}^{1}f(λ_{n}t)f(λ_{m}t)t^α(1-t)^βdt=0,\quad α>-1-2ν, β>-1, \end{equation*}% when $n\neq m$. Considering all possible functions on this class we are lead to the discovery of a new family of generalized Bessel functions including Bessel and Hyperbessel functions as special cases.

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L. D. Abreu, F. Marcellan, S. Yakubovich. 2007-01-05. Hardy-type theorem for functions orthogonal with respect to their zeros. The Jacobi weight case. https://arxiv.org/abs/math/0701175

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