Volchkov Valerii Vladimirovich

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Total publications: 39
Scientific articles: 39

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Doctor of physico-mathematical sciences (1997)
Speciality: 01.01.01 (Real analysis, complex analysis, and functional analysis)
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.


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.


Graduated from Mathematical Faculty of Donetsk State University in 1965 (department of mathematical analysis and theory of functions). Ph. D. thesis was defended in 1991. D. Sci. thesis was defended in 1997. A list of my works contains more than 60 titles.

In 1975 I was awarded the prize of the European Academy.

Main publications:
  1. Volchkov V. V., “Okonchatelnyi variant lokalnoi teoremy o dvukh radiusakh”, Matem. sb., 186:6 (1995), 15–34  mathnet  mathscinet  zmath
  2. Volchkov V. V., “Reshenie problemy nositelya dlya nekotorykh klassov funktsii”, Matem. sb., 188:9 (1997), 13–30  mathnet  mathscinet  zmath
  3. Volchkov V. V., “O mnozhestvakh in'ektivnosti preobrazovaniya Radona na sferakh”, Izv. RAN, ser. matem., 63:3 (1999), 63–76  mathnet  mathscinet  zmath
  4. Volchkov V. V., “Ekstremalnye zadachi o mnozhestvakh Pompeiyu. II”, Matem. sb., 191:5 (2000), 3–16  mathnet  mathscinet  zmath
  5. Volchkov V. V., “O mnozhestvakh in'ektivnosti preobrazovaniya Pompeiyu”, Matem. sb., 190:11 (1999), 51–66  mathnet  mathscinet  zmath
List of publications on Google Scholar
List of publications on ZentralBlatt

Publications in Math-Net.Ru
1. Morera-type theorems in the hyperbolic disc
V. V. Volchkov, Vit.†V.†Volchkov
Izv. RAN. Ser. Mat., 82:1 (2018),  34–64
2. An Analog of the Brown–Schreiber–Taylor Theorem for Weighted Hyperbolic Shifts
V. V. Volchkov, Vit. V. Volchkov
Mat. Zametki, 103:2 (2018),  172–185
3. A new characterization of holomorphic functions in the unit disk
V. V. Volchkov, Vit. V. Volchkov
Probl. Anal. Issues Anal., 7(25):1 (2018),  134–147
4. Spectral synthesis on the group of conformal automorphisms of the unit disc
V. V. Volchkov, Vit. V. Volchkov
Mat. Sb., 209:1 (2018),  3–36
5. Analogs of the Globevnik Problem on Riemannian Two-Point Homogeneous Spaces
V. V. Volchkov, Vit.†V.†Volchkov
Mat. Zametki, 101:3 (2017),  359–372
6. Spectral analysis on the group of conformal automorphisms of the unit disc
V. V. Volchkov, Vit.†V.†Volchkov
Mat. Sb., 207:7 (2016),  57–80
7. —Ónical injectivity sets of the Radon transform on spheres
V. V. Volchkov, Vit.†V.†Volchkov
Algebra i Analiz, 27:5 (2015),  1–31
8. Description of the kernel of the generalized Minkowski transform on the sphere
V. V. Volchkov, Vit.†V.†Volchkov, I. M. Savost'yanova
Izv. RAN. Ser. Mat., 79:1 (2015),  43–62
9. Local two-radii theorems on the multi-dimensional sphere
V. V. Volchkov, Vit.†V.†Volchkov
Izv. RAN. Ser. Mat., 78:1 (2014),  3–24
10. Spherical means on two-point homogeneous spaces and applications
V. V. Volchkov, Vit.†V.†Volchkov
Izv. RAN. Ser. Mat., 77:2 (2013),  3–34
11. Behaviour at infinity of solutions of twisted convolution equations
V. V. Volchkov, Vit. V. Volchkov
Izv. RAN. Ser. Mat., 76:1 (2012),  85–100
12. On the extension problem for solutions of homogeneous convolution equations
V. V. Volchkov, Vit.†V.†Volchkov
Izv. RAN. Ser. Mat., 75:3 (2011),  65–96
13. On a problem of Berenstein–Gay and its generalizations
V. V. Volchkov, Vit.†V.†Volchkov
Izv. RAN. Ser. Mat., 74:4 (2010),  33–62
14. Sets with the Pompeiu Property on the Plane and on the Sphere
V. V. Volchkov, Vit.†V.†Volchkov
Mat. Zametki, 87:1 (2010),  69–82
15. Extremal problems related to the John uniqueness theorem
V. V. Volchkov, Vit. V. Volchkov
Algebra i Analiz, 21:5 (2009),  37–69
16. Convolution equations in many-dimensional domains and on the Heisenberg reduced group
V. V. Volchkov, Vit.†V.†Volchkov
Mat. Sb., 199:8 (2008),  29–60
17. Local two-radii theorem in symmetric spaces
V. V. Volchkov
Mat. Sb., 198:11 (2007),  21–46
18. Uniqueness theorems for solutions of the convolution equation on symmetric spaces
V. V. Volchkov
Izv. RAN. Ser. Mat., 70:6 (2006),  3–18
19. A local two-radii theorem for quasianalytic classes of functions
V. V. Volchkov
Mat. Zametki, 80:4 (2006),  490–500
20. A definitive version of the local two-radii theorem on hyperbolic spaces
V. V. Volchkov
Izv. RAN. Ser. Mat., 65:2 (2001),  3–26
21. Theorems on ball mean values in symmetric spaces
V. V. Volchkov
Mat. Sb., 192:9 (2001),  17–38
22. Extremal problems on Pompeiu sets. II
V. V. Volchkov
Mat. Sb., 191:5 (2000),  3–16
23. Injectivity sets for the Radon transform over a sphere
V. V. Volchkov
Izv. RAN. Ser. Mat., 63:3 (1999),  63–76
24. Injectivity sets of the Pompeiu transform
V. V. Volchkov
Mat. Sb., 190:11 (1999),  51–66
25. Extremal problems on Pompeiu sets
V. V. Volchkov
Mat. Sb., 189:7 (1998),  3–22
26. Uniqueness theorems for some classes of functions with zero spherical means
V. V. Volchkov
Mat. Zametki, 62:1 (1997),  59–65
27. Solution of the support problem for several function classes
V. V. Volchkov
Mat. Sb., 188:9 (1997),  13–30
28. An extremum problem related to Morera's theorem
V. V. Volchkov
Mat. Zametki, 60:6 (1996),  804–809
29. Extremal cases of the Pompeiu problem
V. V. Volchkov
Mat. Zametki, 59:5 (1996),  671–680
30. The final version of the mean value theorem for harmonic functions
V. V. Volchkov
Mat. Zametki, 59:3 (1996),  351–358
31. A definitive version of the local two-radii theorem
V. V. Volchkov
Mat. Sb., 186:6 (1995),  15–34
32. Theorems on two radii on bounded domains of Euclidean spaces
V. V. Volchkov
Differ. Uravn., 30:10 (1994),  1719–1724
33. New two-radii theorems in the theory of harmonic functions
V. V. Volchkov
Izv. RAN. Ser. Mat., 58:1 (1994),  182–194
34. New mean value theorems for polyanalytic functions
V. V. Volchkov
Mat. Zametki, 56:3 (1994),  20–28
35. Mean value theorems for a class of polynomials
V. V. Volchkov
Sibirsk. Mat. Zh., 35:4 (1994),  737–745
36. Morera-type theorems in domains with the weak cone condition
V. V. Volchkov
Izv. Vyssh. Uchebn. Zaved. Mat., 1993, no. 10,  15–20
37. On a problem of Zalcman and its generalizations
V. V. Volchkov
Mat. Zametki, 53:2 (1993),  30–36
38. New theorems on the mean for solutions of the Helmholtz equation
V. V. Volchkov
Mat. Sb., 184:7 (1993),  71–78
39. Uniqueness theorems for multiple lacunary trigonometric series
V. V. Volchkov
Mat. Zametki, 51:6 (1992),  27–31

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