Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)

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Krasnoschekikh G. V., Volchkov V. V. A uniqueness theorem for mean periodic functions on the Bessel – Kingmann hypergroup. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2025, vol. 25, iss. 1, pp. 24-33. DOI: 10.18500/1816-9791-2025-25-1-24-33, EDN: CEYQRP

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Published online: 
28.02.2025
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Language: 
English
Heading: 
Mathematics
Article type: 
Article
UDC: 
517.44
DOI: 
10.18500/1816-9791-2025-25-1-24-33
EDN: 
CEYQRP

A uniqueness theorem for mean periodic functions on the Bessel – Kingmann hypergroup

Autors: 
Krasnoschekikh Gleb Vitalievich, Donetsk State University
Volchkov Vitaliy Vladimirovich, Donetsk State University
Abstract: 

One of the properties of a periodic function on the real axis is that it is completely determined by its values on the period. This fact admits the following nontrivial multidimensional generalization: if a function f∈C∞(Rn) (n≥2) with zero integrals over all spheres (or balls) of fixed radius r is zero in some ball of radius r, then f is zero in Rn. The condition of infinite smoothness of the function f in this statement cannot be relaxed. In this paper, we study a similar phenomenon for solutions of convolution equations related to the generalized Bessel shift operator. First, we consider the case when the convolution factor in the equation is an indicator of a segment symmetric with respect to zero. It is shown that the solutions to such an equation are determined by their values on the specified segment. Further, a generalization of  this property for the general Bessel convolution equation is given. The results obtained are analogues of the well-known uniqueness theorems for mean periodic functions belonging to F. John, Yu.I. Lyubich and A.F. Leontiev.

Key words: 
generalized translation
convolution equations
Bessel functions
spherical transform
Titchmarsh convolution theorem
Acknowledgments: 
The study was conducted on the topic of the state task (registration number: 124012400352-6).
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Received: 
28.09.2023
Accepted: 
13.03.2024
Published: 
28.02.2025
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