Geodesic
On the completeness of a system of Bessel functions of index −3/2 in weighted L2-space
Filomat, Tome 37 (2023) no. 19, p. 6335

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In this paper, we study an integral representation of some class E 2,+ of even entire functions of exponential type σ ≤ 1. We also obtain an analog of the Paley-Wiener theorem related to the class E 2,+. In addition, we find necessary and sufficient conditions for the completeness of a system s k √ xs k J −3/2 (xs k) : k ∈ N in the space L 2 ((0; 1); x 2 dx), where J −3/2 be the Bessel function of the first kind of index −3/2, (s k) k∈N be a sequence of distinct nonzero complex numbers and L 2 ((0; 1); x 2 dx) be the weighted Lebesgue space of all measurable functions f : (0; 1) → C satisfying 1 0 x 2 | f (x)| 2 dx +∞. Those results are formulated in terms of sequences of zeros of functions from the class E 2,+. We also obtain some other sufficient conditions for the completeness of the considered system of Bessel functions. Our results complement similar results on completeness of the systems of Bessel functions of index ν −1, ν Z.
DOI : 10.2298/FIL2319335K
Classification : 33C10, 42C30, 44A15, 30B60, 30D20
Keywords: Bessel function, Paley-Wiener theorem, Phragmén-Lindelöf theorem, Hahn-Banach theorem, Jensen’s formula, Entire function of exponential type, Complete system 
Ruslan Khats. On the completeness of a system of Bessel functions of index −3/2 in weighted L2-space. Filomat, Tome 37 (2023) no. 19, p. 6335 . doi: 10.2298/FIL2319335K
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     author = {Ruslan Khats},
     title = {On the completeness of a system of {Bessel} functions of index \ensuremath{-}3/2 in weighted {L2-space}},
     journal = {Filomat},
     pages = {6335 },
     year = {2023},
     volume = {37},
     number = {19},
     doi = {10.2298/FIL2319335K},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.2298/FIL2319335K/}
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