Completeness conditions of systems of Bessel functions in weighted L 2 -spaces in terms of entire functions

Let Jν be the Bessel function of the first kind of index ν ≥ 1/2, p ∈ R and (ρk)k∈N be a sequence of distinctnonzero complex numbers. Sufficient conditions for the completeness of the system {x−p−1√xρkJν(xρk) : k ∈ N}inthe weighted space L2((0; 1); x2pdx) are found in terms of an entire function with the set of zeros coinciding with thesequence (ρk)k∈N .

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[1] Abreu LD. Completeness, special functions and uncertainty principles over q-linear grids. Journal of Physics A: Mathematical and General 2006; 39 (47): 14567-14580. doi: 10.1088/0305-4470/39/47/004
[2] Boas RP, Pollard H. Complete sets of Bessel and Legendre functions. Annals of Mathematics 1947; 48 (2): 366-384. doi: 10.2307/1969177
[3] Hochstadt H. The mean convergence of Fourier-Bessel series. SIAM Review 1967; 9: 211-218. doi: 10.1137/1009034
[4] Khats’ RV. On conditions of the completeness of some systems of Bessel functions in the space L 2 ((0; 1); x 2p dx). Azerbaijan Journal of Mathematics 2021; 11 (1): 3-10.
[5] Korenev BG. Bessel Functions and their Applications. London, UK: Taylor Francis, Inc., 2002.
[6] Levin BYa. Lectures on Entire Functions. Translations of Mathematical Monographs. Providence, RI, USA: American Mathematical Society, 1996.
[7] Vladimirov VS. Equations of Mathematical Physics. Moscow, Russia: Nauka, 1981. (in Russian)
[8] Vynnyts’kyi BV, Khats’ RV. Some approximation properties of the systems of Bessel functions of index −3/2. Matematychni Studii 2010; 34 (2): 152-159.
[9] Vynnyts’kyi BV, Khats’ RV. Completeness and minimality of systems of Bessel functions. Ufa Mathematical Journal 2013; 5 (2): 131-141. doi: 10.13108/2013-5-2-131
[10] Vynnyts’kyi BV, Khats’ RV. A remark on basis property of systems of Bessel and Mittag-Leffler type functions. Journal of Contemporary Mathematical Analysis 2015; 50 (6): 300-305. doi: 10.3103/S1068362315060060
[11] Vynnyts’kyi BV, Khats’ RV. On the completeness and minimality of sets of Bessel functions in weighted L 2 -spaces. Eurasian Mathematical Journal 2015; 6 (1): 123-131.
[12] Vynnyts’kyi BV, Khats’ RV. Complete biorthogonal systems of Bessel functions. Matematychni Studii 2017; 48 (2): 150-155. doi:10.15330/ms.48.2.150-155
[13] Vynnyts’kyi BV, Khats’ RV, Sheparovych IB. Unconditional bases of systems of Bessel functions. Eurasian Mathematical Journal 2020; 11 (4): 76-86. doi: 10.32523/2077-9879-2020-11-4-76-86