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Various issues related to restrictions on radii in meanvalue formulas are well-known in the theory of harmonic functions. In particular, using the Brown-Schreiber-Taylor theorem on spectral synthesis for motion-invariant subspaces in 𝐶(R^𝑛), one can obtain the following strengthening of the classical mean-value theorem for harmonic functions: if a continuous function on R^𝑛 satisfies the mean-value equations for all balls and spheres of a fixed radius 𝑟, then it is harmonic on R^𝑛. In connection with this result, the following problem arises: recover the Laplacian from the deviation of a function from its average values on balls and spheres of a fixed radius. The aim of this work is to solve this problem. The article uses methods of harmonic analysis, as well as the theory of entire and special functions. The key step in the proof of the main result is expansion of the Dirac delta function in terms of a system of radial distributions supported in a fixed ball, biorthogonal to some system of spherical functions. A similar approach can be used to invert a number of convolution operators with compactly supported radial distributions.

The article for the first time studies the approximation of a function together with its derivatives on the real line by solutions of a multidimensional convolution equation of the form 𝑔 * 𝑇 = 0, where 𝑇 is a given compactly supported radial distribution other than a constant multiplied by the Dirac delta function at zero. The following analogue of the well-known Carleman’s theorem on tangent approximation by entire functions is obtained: if 𝑓 ∈ 𝐶^𝑘(ℝ) for some 𝑘 ∈ ℤ_+ and for any 𝜈 ∈ {0, . . . , 𝑘} there is a finite limit lim 𝑡→+0 (︂𝑑/𝑑𝑡)︂^𝜈 𝑓(ln𝑡), then for an arbitrary positive continuous function 𝜀(𝑡) on ℝ there exists an entire function Φ: ℂ^𝑛 → ℂ, such that Φ|_(ℝ^n) * 𝑇 = 0 and ⃒𝑓^(𝜈)(𝑡) −(︂∂/∂𝑥1)︂^𝜈 Φ(𝑡, 0, . . . , 0)|< 𝜀(𝑒^𝑡) for all 𝑡 ∈ ℝ, 𝜈 ∈ {0, . . . , 𝑘} (Theorem 1). It is shown that when approximating a function on subsets of the real line that do not contain a neighborhood of the point −∞, the condition for the existence of a limit in Theorem 1 can be omitted. In addition, the method of proving Theorem 1 allows one to obtain new results that are of interest for the theory of convolution equations of the indicated type. These are results about the growth of solutions (Corollary 3), on the distribution of values (Theorem 2), and also on the solvability of the interpolation problem for solutions of the convolution equation (Corollary 4).

Let 𝑛 >= 2, 𝑉_𝑟(R^𝑛) be the set of functions with zero integrals over all balls in R^𝑛 of radius 𝑟. Various interpolation problems for the class 𝑉_𝑟(R^𝑛) are studied. In the case when the set of interpolation nodes is finite, we solve the interpolation problem under general conditions. For the problems with infinite set of nodes, some sufficient conditions of solvability are founded. Note that an essential condition is that the definition of the class 𝑉_𝑟(R^𝑛) involves integration over balls. For instance, it can be shown that the analogues of our results in which the class of functions is defined using zero integrals over all shifts of a fixed parallelepiped in R^𝑛 do not hold true.

This paper is devoted to a study of the following version of the mean periodic extension problem: (i) Suppose that 𝑇 ∈ ℰ′ (R^𝑛), 𝑛 ≥ 2, and 𝐸 is a non-empty subset of R^𝑛. Let 𝑓 ∈ 𝐶(𝐸). What conditions guarantee that there is an 𝐹 ∈ 𝐶(R^𝑛) coinciding with 𝑓 on 𝐸, such that 𝐹 *𝑇 = 0 in R^𝑛 ? (ii) If such an extension 𝐹 exists, then estimate the growth of 𝐹 at infinity. In this paper, we present a solution of this problem for a broad class of distributions 𝑇 in the case when 𝐸 is a segment in R^n.