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Chebyshevskii Sbornik, 2023, Volume 24, Issue 2, Pages 63–80
DOI: https://doi.org/10.22405/2226-8383-2023-24-2-63-80
(Mi cheb1308)
 

The problem of finding a function by its ball means values

N. P. Volchkovaa, Vit. V. Volchkovb

a Donetsk National Technical University (Donetsk)
b Donetsk State University (Donetsk)
References:
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 $f:\mathbb{R}^n\to \mathbb{C}$ with nonzero integrals over all balls of radii $r_1$ and $r_2$ in $\mathbb{R}^n$ it is necessary and sufficient that $r_1/r_2\in E_n$, where $E_n$ is the set of all possible ratios of positive zeros of the Bessel function $J_{n/2}$. The condition $r_1/r_2\notin E_n$is equivalent to the equality $\mathcal{Z}_{+}\big(\widetilde{\chi}_{r_1}\big)\cap\mathcal{Z}_{+}\big(\widetilde{\chi}_{r_2}\big)=\varnothing$, where $\chi_{r}$ is the indicator of the ball $B_r=\{x\in\mathbb{R}^n: |x|<r\}$, $\widetilde{\chi}_{r}$ is the spherical transform (Fourier-Bessel transform) of the indicator $\chi_{r}$, $\mathcal{Z}_{+}(\widetilde{\chi}_{r})$ is the set of all positive zeros of even entire function $\widetilde{\chi}_{r}$. In terms of convolutions, L. Zalcman's theorem means that the operator
$$\mathcal{P}f=(f\ast \chi_{r_1}, f\ast \chi_{r_2}), f\in L^{1,\mathrm{loc}}(\mathbb{R}^n) $$
is injective if and only if $r_1/r_2\notin E_n$. In this paper, a new formula for the inversion of the operator $\mathcal{P}$ is found under the condition $r_1/r_2\notin E_n$. The result obtained significantly simplifies the previously known procedures for recovering a function $f$ from given ball means values $f\ast \chi_{r_1}$ и $f\ast \chi_{r_2}$. The proofs use the methods of harmonic analysis, as well as the theory of entire and special functions.
Keywords: mean periodic functions, radial distributions, two-radii theorem, inversion formulas.
Received: 04.08.2022
Accepted: 14.06.2023
Document Type: Article
UDC: 3517.5
Language: Russian
Citation: N. P. Volchkova, Vit. V. Volchkov, “The problem of finding a function by its ball means values”, Chebyshevskii Sb., 24:2 (2023), 63–80
Citation in format AMSBIB
\Bibitem{VolVol23}
\by N.~P.~Volchkova, Vit.~V.~Volchkov
\paper The problem of finding a function by its ball means values
\jour Chebyshevskii Sb.
\yr 2023
\vol 24
\issue 2
\pages 63--80
\mathnet{http://mi.mathnet.ru/cheb1308}
\crossref{https://doi.org/10.22405/2226-8383-2023-24-2-63-80}
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  • https://www.mathnet.ru/eng/cheb/v24/i2/p63
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