The problem of finding a function by its ball means values
https://doi.org/10.22405/2226-8383-2023-24-2-63-80
Abstract
A classical property of a non-constant 2𝑟-periodic function on the real axis is that it has no period incommensurable with 𝑟. One of the multidimensional analogues of this statement is the following well-known theorem of L. Zalcman on two radii: for the existence of a nonzero locally summable function 𝑓 : R𝑛 → C with nonzero integrals over all balls of radii 𝑟1 and 𝑟2 in R𝑛 it is necessary and sufficient that 𝑟1/𝑟2 ∈ 𝐸𝑛, where 𝐸𝑛 is the set of all possible ratios of positive zeros of the Bessel function 𝐽𝑛/2. The condition 𝑟1/𝑟2 /∈ 𝐸𝑛is equivalent to the equality 𝒵+(︀̃︀𝜒𝑟1)︀∩ 𝒵+(︀̃︀𝜒𝑟2)︀= ∅, where 𝜒𝑟 is the indicator of the ball 𝐵𝑟 = {𝑥 ∈ R𝑛 : |𝑥| < 𝑟}, ̃︀𝜒𝑟 is the spherical transform (Fourier-Bessel transform) of the indicator 𝜒𝑟, 𝒵+(̃︀𝜒𝑟) is the set of all positive zeros of even entire function ̃︀𝜒𝑟. In terms of convolutions, L. Zalcman’s theorem means that the operator
$$𝒫𝑓 = (𝑓 * 𝜒𝑟1 , 𝑓 * 𝜒𝑟2 ), 𝑓 ∈ 𝐿^(1,loc) (R𝑛)$$
is injective if and only if 𝑟1/𝑟2 /∈ 𝐸𝑛. In this paper, a new formula for the inversion of the operator 𝒫 is found under the condition 𝑟1/𝑟2 /∈ 𝐸𝑛. The result obtained significantly simplifies the previously known procedures for recovering a function 𝑓 from given ball means values 𝑓 *𝜒𝑟1 и 𝑓 * 𝜒𝑟2 . The proofs use the methods of harmonic analysis, as well as the theory of entire and special functions.
About the Authors
Natalia Petrovna Volchkova
Donetsk National Technical University
Ukraine
Ukraine
candidate of physical and mathematical sciences, associate
professor
Vitaliy Vladimirovich Volchkov
Donetsk State University
Ukraine
Ukraine
doctor of physical and mathematical sciences, professor
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For citations:
Volchkova N.P., Volchkov V.V. The problem of finding a function by its ball means values. Chebyshevskii Sbornik. 2023;24(2):63-80. (In Russ.) https://doi.org/10.22405/2226-8383-2023-24-2-63-80
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