Completeness of the systems of Bessel functions of index $-5/2$

Authors

https://doi.org/10.15330/cmp.16.1.93-102

Keywords:

Bessel function, Paley-Wiener theorem, Phragmén-Lindelöf theorem, Fubini's theorem, Hurwitz's theorem, Hahn-Banach theorem, Jensen's formula, entire function of exponential type, complete system
Published online: 2024-05-13

Abstract

Let $L^2((0;1);x^4 dx)$ be the weighted Lebesgue space of all measurable functions $f:(0;1)\rightarrow\mathbb C$, satisfying $\int_{0}^1 t^4 |f(t)|^2\, dt<+\infty$. Let $J_{-5/2}$ be the Bessel function of the first kind of index $-5/2$ and $(\rho_k)_{k\in\mathbb N}$ be a sequence of distinct nonzero complex numbers. Necessary and sufficient conditions for the completeness of the system $\big\{\rho_k^2\sqrt{x\rho_k}J_{-5/2}(x\rho_k):k\in\mathbb N\big\}$ in the space $L^2((0;1);x^4 dx)$ are found in terms of an entire function with the set of zeros coinciding with the sequence $(\rho_k)_{k\in\mathbb N}$. In this case, we study an integral representation of some class $E_{4,+}$ of even entire functions of exponential type $\sigma\le 1$. This complements similar results on approximation properties of the systems of Bessel functions of negative half-integer index less than $-1$, due to B. Vynnyts'kyi, V. Dilnyi, O. Shavala and the author.

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How to Cite
(1)
Khats', R. Completeness of the Systems of Bessel Functions of Index $-5 2$. Carpathian Math. Publ. 2024, 16, 93-102.