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