Speciality:
01.01.01 (Real analysis, complex analysis, and functional analysis)
E-mail: Website: https://www.mathnet.ru/rus/rm10221 Keywords: theory of numbers,
integral geometry,
harmonic analysis,
convolutional equations,
mean periodicity,
the problem of continuation,
the Fourier transform,
Heisenberg group,
twisted convolution equation,
two-radii theorems,
confluent hypergeometric functions.
Subject:
Various classes of functions with zero integrals over all balls of a fixed radius are studied. For functions in such classes a description in the form of a series in special functions is obtained and a uniqueness theorem is proved. There results make it possible to solve completely the problem of existence of a non-trivial function with zero integrals over balls of radius assuming either of two given values. Also there results make it possible to solve completely the support problem for several classes of functions with zero ball averages. As consequence, a far-reaching generalization of the well-known Zalcman's two-radii theorem is obtained. A solution of the local Pompeiu problem is obtained for functions with vanishing integrals over parallelepipeds lying in a fixed ball. A solution of the problem of describing the kernel of the Radon transform over a sphere with respect to sets with spherical symmetry is obtained. The solution enabled us, in particular, to characterize all injectivity sets of this type. The technique used in the proofs enabled us to obtain other exact results concerning spherical means, namely, new two-radii theorems and a uniqueness theorem.
Main publications:
Volchkov V. V., “Okonchatelnyi variant lokalnoi teoremy o dvukh radiusakh”, Matem. sb., 186:6 (1995), 15–34
Volchkov V. V., “Reshenie problemy nositelya dlya nekotorykh klassov funktsii”, Matem. sb., 188:9 (1997), 13–30
Volchkov V. V., “O mnozhestvakh in'ektivnosti preobrazovaniya Radona na sferakh”, Izv. RAN, ser. matem., 63:3 (1999), 63–76
Volchkov V. V., “Ekstremalnye zadachi o mnozhestvakh Pompeiyu. II”, Matem. sb., 191:5 (2000), 3–16
