# Volchkov V. V.

### Analogs of the spherical transform on the hyperbolic plane

Vasilyanskaya V. S., Volchkov V. V.

↓ Abstract

Ukr. Mat. Zh. - 2016. - 68, № 4. - pp. 469-484

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

### One Problem Connected with the Helgason Support Problem

Savost’yanova I. M., Volchkov V. V., Volchkov V. V.

↓ Abstract

Ukr. Mat. Zh. - 2015. - 67, № 9. - pp. 1189-1200

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.

### Smoothing of the Singularities of Functions Whose Integrals over the Balls on a Sphere are Zero

Savost’yanova I. M., Volchkov V. V.

Ukr. Mat. Zh. - 2015. - 67, № 2. - pp. 272-278

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.

### On One Minkowski–Radon Problem and Its Generalizations

Savost’yanova I. M., Volchkov V. V.

Ukr. Mat. Zh. - 2014. - 66, № 10. - pp. 1332–1347

We study functions on a sphere with zero weighted means over the circles of fixed radius. A description of these functions is obtained in the form of series in special functions.

### Analog of the John theorem for weighted spherical means on a sphere

Savost’yanova I. M., Volchkov V. V.

Ukr. Mat. Zh. - 2013. - 65, № 5. - pp. 611–619

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.

### New Integral Representations for a Hypergeometric Function

Volchkov V. V., Volchkov V. V.

Ukr. Mat. Zh. - 2001. - 53, № 4. - pp. 435-440

We obtain new integral representations for a hypergeometric function.

### On Multipliers in Hardy Spaces

Ukr. Mat. Zh. - 2001. - 53, № 3. - pp. 417-421

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

### On the Pompeiu problem and its generalizations

Ukr. Mat. Zh. - 1993. - 45, № 10. - pp. 1444–1448

Functions are investigated whose integrals over a given collection of sets are zero. Pompeiu sets are described in terms of the approximation of their indicators by linear combinations of the indicators of balls with special radii.

### Functions with zero integrals over cubes

Ukr. Mat. Zh. - 1991. - 43, № 6. - pp. 723–727