ON ONE ZALCMAN PROBLEM FOR THE MEAN VALUE OPERATOR

Abstract

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.