Search

Let $\mathbb{R}^n$ be a Euclidean space of dimension $n\geq 2$ . For a domain $G\subset \mathbb{R}^n$ , we denote by $V_r(G)$ the set of functions $f\in L_{\mathrm{loc}}(G)$ having zero integrals over all closed balls of radius r contained in G (if domain G does not contain such balls, we set $V_r(G)=L_{\mathrm{loc}}(G)$ ). Let E be a nonempty subset of $\mathbb{R}^n$ . In this paper we study the following questions related to the extension problem. 1) Which conditions guarantee the extension of a continuous function defined on E to a continuous function of class $V_r(\mathbb{R}^n)$ defined on the whole $\mathbb{R}^n$ ? 2) If the above extension exists, obtain growth estimates of the extended function at infinity. Theorem 1 of this paper shows that for a wide class of continuous functions on segment E defined in terms of the modulus of continuity, there exists an extension to a bounded function of class $(V_r\cap C)(\mathbb{R}^n)$ regardless of the length of segment E. A similar result is not true for open sets E with a diameter greater than 2r, even without conditions for extension growth. Theorem 1 also contains an estimate of the velocity decrease of the extended function at infinity in directions orthogonal to the segment E.

We study the following version of the mean periodic extension problem. (i) Suppose that $$T \in \mathcal{E}'({{\mathbb{R}}^{n}})$$ , n ≥ 2, and E is a nonempty closed subset of $${{\mathbb{R}}^{n}}$$ . What conditions guarantee that, for a function f ∈ C(E), there is a function $$F \in C({{\mathbb{R}}^{n}})$$ coinciding with f on E such that $$F * T = 0$$ in $${{\mathbb{R}}^{n}}$$ ? (ii) If such an extension F exists, then estimate the growth of F at infinity. We present a solution of this problem for a broad class of distributions T in the case when E is an interval in $${{\mathbb{R}}^{n}}$$ .

Let $\Gamma$ be a closed smooth Jordan curve in the complex plane $\mathbb{C}$ ,G be a bounded domain in $\mathbb{C}$ with the boundary $\Gamma$ , and let $\overline{G}=G\cup\Gamma$ . We study functions that are continuous in $\mathbb{C}\setminus G$ and harmonic in $\mathbb{C}\setminus\overline{G}$ that grow more slowly than the function $|z|^2$ at $z\to\infty$ . It is shown that, if in the class of such functions there exists a solution to the overdetermined Neumann boundary value problem in which the function equals zero on $\Gamma$ and $\mu$ -almost everywhere on $\Gamma$ there exists and equals one the normal derivative of this function, then the domain G is a disk (Theorem 1). In this case, the solution is unique and it coincides up to a constant with the fundamental solution for the Laplace operator in $\mathbb{C}$ with a singularity in the center of the disk G. The proof of Theorem 1 is based on the application of the conformal mapping of the exterior of the unit disk onto the domain $\mathbb{C}\setminus \overline{G}$ . This mapping allows us to reduce the original problem for the domain $\mathbb{C}\setminus \overline{G}$ to overdetermined boundary value problem for the exterior of the disk in which the main difficulty is the heterogeneity of the boundary condition for the normal derivative. To study this condition some subtle results were required on the boundary properties of a function that performs the indicated conformal mapping as well as some properties of the Hardy classes Hp in the unit disk. Theorem 2 of the paper shows that in the general case the conditions in Theorem 1 cannot be relaxed. It states the existence of a bounded domain $G\subset\mathbb{C}$ different from a disk with a smooth Jordan boundary $\Gamma$ and functions $f_1,f_2,f_3\in C(\mathbb{C}\setminus G)$ harmonic in $\mathbb{C}\setminus\overline{G}$ for each of which exactly one of the conditions of Theorem 1 is not satisfied.

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.

Let $${{V}_{r}}({{\mathbb{R}}^{n}})$$, n ≥ 2, be the set of functions $$f \in {{L}_{{{\text{loc}}}}}({{\mathbb{R}}^{n}})$$ with zero integrals over all balls in $${{\mathbb{R}}^{n}}$$ of radius r. Various interpolation problems for the class $${{V}_{r}}({{\mathbb{R}}^{n}})$$ are studied. In the case when the set of interpolation nodes is finite, the multiple interpolation problem is solved under general assumptions. For problems with an infinite set of nodes, sufficient solvability conditions are founded. Additionally, we construct a new example of a subset in $${{\mathbb{R}}^{n}}$$ for which some nontrivial real analytic function of the class $${{V}_{r}}({{\mathbb{R}}^{n}})$$ vanishes.

We study functions given in the form of a series in Bessel's functions of the first kind. The admissible asymptotic behavior of such functions at infinity is founded. As a consequence we obtain an analog of Liouville's theorem for the Fourier-Bessel and Dini developments.

We consider precise conditions ensuring that a function having zero integrals over all balls of fixed radius is equal to zero. The case where, together with zero integrals for f over congruent balls, the functions xjf for all j have zero integrals over these balls is completely investigated. An inversion formula recovering a function in the class C∞ from indicated integral moments is derived.

In the present paper, using a development of the technique of transmutation mappings, we obtain the first weighted analog of the well-known Brown–Schreiber–Taylor theorem on the eigenvalues of the Laplacian for corresponding spaces of continuous functio

We study functions on a sphere with a pricked point having zero integrals with a given weight over all admissible “hemispheres”. We find a condition under which the point is a removable set for such a class of functions. We show that this condition cannot be dropped or substantially weakened.

We study the functions on the punctured n-dimensional sphere having zero integrals over all admissible “hemispheres.” We find a condition for the point to be a removable set for this class of functions and show that the condition cannot be dropped or substantially improved.

A non-Euclidean analog of the generalized Darboux equation is considered. For the case where its solutions are radial functions of second variable we obtain a uniqueness result (Theorem 1) which deals with zero sets of these solutions. The example of the function in Theorem 2 of the paper shows that Theorem 1 cannot be essentially reinforced.

On a two-point homogeneous space X, we consider the problem of describing the set of continuous functions having zero integrals over all spheres enclosing the given ball. We obtain the solution of this problem and its generalizations for an annular domain in X. By way of applications, we prove new uniqueness theorems for functions with zero spherical means.

We introduce the notion of “s”-convolution in the hyperbolic plane $$ {\mathrm{\mathbb{H}}}^2 $$ and consider its properties. Analogs of the Helgason spherical transform in the spaces of compactly supported distributions in $$ {\mathrm{\mathbb{H}}}^2 $$ are investigated. We prove a Paley–Wiener–Schwartz-type theorem for the indicated transforms.

We solve the problem of description of the set of continuous functions in annular subdomains of the n-dimensional sphere with zero integrals over all (n − 1)-dimensional spheres covering the inner spherical cap. As an application, we establish a spherical analog of the Helgason support theorem and new uniqueness theorems for functions with zero spherical means.

We study functions defined on a sphere with prickled point whose integrals over all admissible “hemispheres” are equal to zero. A condition is established under which the point is a removable set for this class of functions. It is shown that this condition cannot be omitted or noticeably weakened.

We study generalizations of the class of functions with zero integrals over the balls of fixed radius. An analog of the John uniqueness theorem is obtained for weighted spherical means on a sphere.

We obtain new sufficient conditions under which a set on the plane has the Pompeiu property. This result allows us to construct first examples of domains with the Pompeiu property with non-Lipschitz (and even fractal) boundary. In addition, we study the problem of determining the least radius of the ball on the sphere in which a given set is a Pompeiu set. We obtain the solution of this problem in the case of a biangle and a spherical half-disk. We also consider some applications to questions of complex analysis.

In this paper, we present new uniqueness results related to geometric aspects of mean periodicity on various homogeneous spaces. Among these results, we point out the equivalence of the local and the global Pompeiu property for arbitrary families of compactly supported distributions, the solution of the local Pompeiu problem for the class of non real-analytic functions, and local versions of the two-radii theorem on symmetric spaces of arbitrary rank.

Various classes of functions on a non-compact rank-one Riemannian symmetric space X with vanishing integrals over all balls of fixed radius are studied. A description in the form of a series in hypergeometric functions is obtained for such classes and a uniqueness theorem is proved. This makes it possible to establish the local two-radii theorem in X in a definitive form. Bibliography: 45 titles.

The problem of the description of the class of functions with zero integral over each ball of prescribed radius lying in a fixed ball is solved for compact two-point homogeneous spaces.Bibliography: 35 titles.

We obtain a description of the kernel of the local Pompeiu transform for some families of distributions on the sphere.

An exact characteristic of maximal smoothness of nontrivial functions with zero integrals over balls whose radii belong to a given two-element set is obtained..

New uniqueness theorems for periodic (in mean) functions on quaternion hyperbolic space are obtained.

New uniqueness theorems are obtained for mean periodic functions on complex hyperbolic spaces.

We investigate the accuracy of certain sufficient conditions for multipliers of power series in Hardy spaces.

The class of functions with zero integrals over all balls of one fixed radius on the complex hyperbolic space is described.

We obtain conditions for the equivalence of certain differential operators in spaces of analytic functions over the Tate field.

Let ϕ be a distribution with compact support in ℝ n , n ⩾ 2. For fixed λ ∊ M(n) we define the distribution λϕ acting in ɛ(ℝ n ) by the formula $$ \left\langle {\lambda \phi ,f\left( x \right)} \right\rangle = \left\langle {\phi ,f\left( {\lambda ^{ - 1} x} \right)} \right\rangle , f \in \mathcal{E}\left( {\mathbb{R}^n } \right). $$ Let \( \mathcal{F} = \left\{ {\phi _1 , \ldots ,\phi _m } \right\} \) be a given collection of nonzero distributions of ɛ′(ℝ n ). For an open subset \( \mathcal{U} \) of ℝ n such that each of sets $$ \mathfrak{X}_j = \left\{ {\lambda \in M\left( n \right):supp \lambda \phi _j \subset \mathcal{U}} \right\}, j = 1, \ldots ,m $$ (7.1) is non-empty the Pompeiu transform \( \mathcal{P}_\mathcal{F} \) maps \( \mathcal{E}\left( \mathcal{U} \right) \) into \( \mathcal{E}\left( {\mathfrak{X}_1 } \right) \times \cdots \times \mathcal{E}\left( {\mathfrak{X}_m } \right) \) in accordance with the formula $$ \mathcal{P}_\mathcal{F} f = \left( {f_1 , \ldots ,f_m } \right), f \in \mathcal{E}\left( \mathcal{U} \right), $$ where \( f_j \left( \lambda \right) = \left\langle {\lambda \phi _j ,f} \right\rangle ,\lambda \in \mathfrak{X}_j ,j = 1, \ldots ,m \)