ON ONE ZALCMAN PROBLEM FOR THE MEAN VALUE OPERATOR

Abstract

Let D′(Rn) and E′(Rn) be the spaces of distributions and compactly supported distributions on Rn, n≥2, respectively, let E′♮(Rn) be the space of all radial (invariant under rotations of the space Rn) distributions in E′(Rn), let T˜ be the spherical transform (Fourier--Bessel transform) of a distribution T∈E′♮(Rn), and let Z+(T˜) be the set of all zeros of an even entire function T˜ lying in the half-plane Rez≥0 and not belonging to the negative part of the imaginary axis. Let σr be the surface delta function concentrated on the sphere Sr={x∈Rn:|x|=r}. The problem of L. Zalcman on reconstructing a distribution f∈D′(Rn) from known convolutions f∗σr1 and f∗σr2 is studied. This problem is correctly posed only under the condition r1/r2∉Mn, where Mn is the set of all possible ratios of positive zeros of the Bessel function Jn/2−1. The paper shows that if r1/r2∉Mn, then an arbitrary distribution f∈D′(Rn) can be expanded into an unconditionally convergent series f=∑λ∈Z+(Ω˜r1)∑μ∈Z+(Ω˜r2)4λμ(λ2−μ2)Ω˜′r1(λ)Ω˜′r2(μ)(Pr2(Δ)((f∗σr2)∗Ωλr1)−Pr1(Δ)((f∗σr1)∗Ωμr2)) in the space D′(Rn), where Δ is the Laplace operator in Rn, Pr is an explicitly given polynomial of degree [(n+5)/4], and Ωr and Ωλr are explicitly constructed radial distributions supported in the ball |x|≤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.