ON ONE ZALCMAN PROBLEM FOR THE MEAN VALUE OPERATOR

Let $\mathcal{D}'(\mathbb{R}^n)$ and $\mathcal{E}'(\mathbb{R}^n)$ be the spaces of distributions and compactly supported distributions on $\mathbb{R}^n$, $n\geq 2$ respectively, let $\mathcal{E}'_{\natural}(\mathbb{R}^n)$ be the space of all radial (invariant under rotations...

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Bibliographic Details
Main Authors:	Natalia P. Volchkova, Vitaliy V. Volchkov
Format:	Article
Language:	English
Published:	Ural Branch of the Russian Academy of Sciences and Ural Federal University named after the first President of Russia B.N.Yeltsin, Krasovskii Institute of Mathematics and Mechanics 2023-07-01
Series:	Ural Mathematical Journal
Subjects:	compactly supported distributions, fourier > bessel transform, two-radii theorem, inversion formulas.
Online Access:	https://umjuran.ru/index.php/umj/article/view/532
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Summary:	Let $\mathcal{D}'(\mathbb{R}^n)$ and $\mathcal{E}'(\mathbb{R}^n)$ be the spaces of distributions and compactly supported distributions on $\mathbb{R}^n$, $n\geq 2$ respectively, let $\mathcal{E}'_{\natural}(\mathbb{R}^n)$ be the space of all radial (invariant under rotations of the space $mathbb{R}^n$) distributions in $\mathcal{E}'(\mathbb{R}^n)$, let $\widetilde{T}$ be the spherical transform (Fourier–Bessel transform) of a distribution $T\in\mathcal{E}'_{\natural}(\mathbb{R}^n)$, and let $\mathcal{Z}_{+}(\widetilde{T})$ be the set of all zeros of an even entire function $\widetilde{T}$ lying in the half-plane $\mathrm{Re} \, z\geq 0$ and not belonging to the negative part of the imaginary axis. Let $\sigma_{r}$ be the surface delta function concentrated on the sphere $S_r=\{x\in\mathbb{R}^n: \|x\|=r\}$. The problem of L. Zalcman on reconstructing a distribution $f\in \mathcal{D}'(\mathbb{R}^n)$ from known convolutions $f\ast \sigma_{r_1}$ and $f\ast \sigma_{r_2}$ is studied. This problem is correctly posed only under the condition $r_1/r_2\notin M_n$, where $M_n$ is the set of all possible ratios of positive zeros of the Bessel function $J_{n/2-1}$. The paper shows that if $r_1/r_2\notin M_n$, then an arbitrary distribution $f\in \mathcal{D}'(\mathbb{R}^n)$ can be expanded into an unconditionally convergent series $$ f=\sum\limits_{\lambda\in\mathcal{Z}_{+}(\widetilde{\Omega}_{r_1})}\,\,\, \sum\limits_{\mu\in\mathcal{Z}_+(\widetilde{\Omega}_{r_2})} \frac{4\lambda\mu}{(\lambda^2-\mu^2) \widetilde{\Omega}_{r_1}^{\,\,\,\displaystyle{'}}(\lambda)\widetilde{\Omega}_{r_2}^{\,\,\,\displaystyle{'}}(\mu)}\Big (P_{r_2} (\Delta) \big((f\ast\sigma_{r_2})\ast \Omega_{r_1}^{\lambda}\big)-P_{r_1} (\Delta) \big((f\ast\sigma_{r_1})\ast \Omega_{r_2}^{\mu}\big)\Big) $$ in the space $\mathcal{D}'(\mathbb{R}^n)$, where $\Delta$ is the Laplace operator in $\mathbb{R}^n$, $P_r$ is an explicitly given polynomial of degree $[(n+5)/4]$, and $\Omega_{r}$ and $\Omega_{r}^{\lambda}$ are explicitly constructed radial distributions supported in the ball $\|x\|\leq r$. The proof uses the methods of harmonic analysis, as well as the theory of entire and special functions. By a similar technique, it is possible to obtain inversion formulas for other convolution operators with radial distributions.
ISSN:	2414-3952

Summary:	Let \(\mathcal{D}'(\mathbb{R}^n)\) and \(\mathcal{E}'(\mathbb{R}^n)\) be the spaces of distributions and compactly supported distributions on \(\mathbb{R}^n\), \(n\geq 2\) respectively, let \(\mathcal{E}'_{\natural}(\mathbb{R}^n)\) be the space of all radial (invariant under rotations of the space \(mathbb{R}^n\)) distributions in \(\mathcal{E}'(\mathbb{R}^n)\), let \(\widetilde{T}\) be the spherical transform (Fourier–Bessel transform) of a distribution \(T\in\mathcal{E}'_{\natural}(\mathbb{R}^n)\), and let \(\mathcal{Z}_{+}(\widetilde{T})\) be the set of all zeros of an even entire function \(\widetilde{T}\) lying in the half-plane \(\mathrm{Re} \, z\geq 0\) and not belonging to the negative part of the imaginary axis. Let \(\sigma_{r}\) be the surface delta function concentrated on the sphere \(S_r=\{x\in\mathbb{R}^n: \|x\|=r\}\). The problem of L. Zalcman on reconstructing a distribution \(f\in \mathcal{D}'(\mathbb{R}^n)\) from known convolutions \(f\ast \sigma_{r_1}\) and \(f\ast \sigma_{r_2}\) is studied. This problem is correctly posed only under the condition \(r_1/r_2\notin M_n\), where \(M_n\) is the set of all possible ratios of positive zeros of the Bessel function \(J_{n/2-1}\). The paper shows that if \(r_1/r_2\notin M_n\), then an arbitrary distribution \(f\in \mathcal{D}'(\mathbb{R}^n)\) can be expanded into an unconditionally convergent series $$ f=\sum\limits_{\lambda\in\mathcal{Z}_{+}(\widetilde{\Omega}_{r_1})}\,\,\, \sum\limits_{\mu\in\mathcal{Z}_+(\widetilde{\Omega}_{r_2})} \frac{4\lambda\mu}{(\lambda^2-\mu^2) \widetilde{\Omega}_{r_1}^{\,\,\,\displaystyle{'}}(\lambda)\widetilde{\Omega}_{r_2}^{\,\,\,\displaystyle{'}}(\mu)}\Big (P_{r_2} (\Delta) \big((f\ast\sigma_{r_2})\ast \Omega_{r_1}^{\lambda}\big)-P_{r_1} (\Delta) \big((f\ast\sigma_{r_1})\ast \Omega_{r_2}^{\mu}\big)\Big) $$ in the space \(\mathcal{D}'(\mathbb{R}^n)\), where \(\Delta\) is the Laplace operator in \(\mathbb{R}^n\), \(P_r\) is an explicitly given polynomial of degree \([(n+5)/4]\), and \(\Omega_{r}\) and \(\Omega_{r}^{\lambda}\) are explicitly constructed radial distributions supported in the ball \(\|x\|\leq r\). The proof uses the methods of harmonic analysis, as well as the theory of entire and special functions. By a similar technique, it is possible to obtain inversion formulas for other convolution operators with radial distributions.
ISSN:	2414-3952

ON ONE ZALCMAN PROBLEM FOR THE MEAN VALUE OPERATOR

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