01.01.01 (Real analysis, complex analysis, and functional analysis)
E-mail:
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.
Biography
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:
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
V. V. Volchkov, Vit. V. Volchkov, “Spectral Synthesis on the Reduced Heisenberg Group”, Mat. Zametki, 113:1 (2023), 46–57; Math. Notes, 113:1 (2023), 49–58
V. V. Volchkov, Vit. V. Volchkov, “Approximation of functions on rays in $\Bbb{R}^n$ by solutions to convolution equations”, Sibirsk. Mat. Zh., 64:1 (2023), 56–64; Siberian Math. J., 64:1 (2023), 48–55
V. V. Volchkov, Vit. V. Volchkov, “Tangent approximation by solutions of the convolution equation”, Probl. Anal. Issues Anal., 11(29):3 (2022), 125–142
2021
4.
V. V. Volchkov, Vit. V. Volchkov, “Continuous mean periodic extension of functions from an interval”, Dokl. RAN. Math. Inf. Proc. Upr., 496 (2021), 21–25; Dokl. Math., 103:1 (2021), 14–18
V. V. Volchkov, Vit. V. Volchkov, “Continuous extension of functions from a segment to functions in $\mathbb{R}^n$ with zero ball means”, Izv. Vyssh. Uchebn. Zaved. Mat., 2021, no. 3, 3–14; Russian Math. (Iz. VUZ), 65:3 (2021), 1–11
V. V. Volchkov, Vit. V. Volchkov, “Interpolation problems for functions with zero integrals over balls of fixed radius”, Dokl. RAN. Math. Inf. Proc. Upr., 490 (2020), 20–23; Dokl. Math., 101:1 (2020), 16–19
V. V. Volchkov, Vit. V. Volchkov, “On the radial symmetry property for harmonic functions”, Izv. Vyssh. Uchebn. Zaved. Mat., 2020, no. 10, 12–23; Russian Math. (Iz. VUZ), 64:10 (2020), 9–19
9.
V. V. Volchkov, Vit. V. Volchkov, “On the problem of mean periodic extension”, Probl. Anal. Issues Anal., 9(27):2 (2020), 138–151
10.
V. V. Volchkov, Vit. V. Volchkov, “Analogs of the liouville property for harmonic functions on unbounded domains”, Sibirsk. Mat. Zh., 61:4 (2020), 752–764; Siberian Math. J., 61:4 (2020), 589–599
11.
V. V. Volchkov, Vit. V. Volchkov, “Overdetermined Neumann boundary value problem in unbounded domains”, Ufimsk. Mat. Zh., 12:2 (2020), 10–20; Ufa Math. J., 12:2 (2020), 10–20
V. V. Volchkov, Vit. V. Volchkov, “Morera-type theorems in the hyperbolic disc”, Izv. RAN. Ser. Mat., 82:1 (2018), 34–64; Izv. Math., 82:1 (2018), 31–60
13.
V. V. Volchkov, Vit. V. Volchkov, “An Analog of the Brown–Schreiber–Taylor Theorem for Weighted Hyperbolic Shifts”, Mat. Zametki, 103:2 (2018), 172–185; Math. Notes, 103:2 (2018), 175–186
14.
V. V. Volchkov, Vit. V. Volchkov, “A new characterization of holomorphic functions in the unit disk”, Probl. Anal. Issues Anal., 7(25):1 (2018), 134–147
15.
V. V. Volchkov, Vit. V. Volchkov, “Spectral synthesis on the group of conformal automorphisms of the unit disc”, Mat. Sb., 209:1 (2018), 3–36; Sb. Math., 209:1 (2018), 1–34
2017
16.
V. V. Volchkov, Vit. V. Volchkov, “Analogs of the Globevnik Problem on Riemannian Two-Point Homogeneous Spaces”, Mat. Zametki, 101:3 (2017), 359–372; Math. Notes, 101:3 (2017), 417–428
2016
17.
V. V. Volchkov, Vit. V. Volchkov, “Spectral analysis on the group of conformal automorphisms of the unit disc”, Mat. Sb., 207:7 (2016), 57–80; Sb. Math., 207:7 (2016), 942–963
V. V. Volchkov, Vit. V. Volchkov, “Ñînical injectivity sets of the Radon transform on spheres”, Algebra i Analiz, 27:5 (2015), 1–31; St. Petersburg Math. J., 27:5 (2016), 709–730
V. V. Volchkov, Vit. V. Volchkov, I. M. Savost'yanova, “Description of the kernel of the generalized Minkowski transform on the sphere”, Izv. RAN. Ser. Mat., 79:1 (2015), 43–62; Izv. Math., 79:1 (2015), 40–59
V. V. Volchkov, Vit. V. Volchkov, “Local two-radii theorems on the multi-dimensional sphere”, Izv. RAN. Ser. Mat., 78:1 (2014), 3–24; Izv. Math., 78:1 (2014), 1–21
V. V. Volchkov, Vit. V. Volchkov, “Spherical means on two-point homogeneous spaces and applications”, Izv. RAN. Ser. Mat., 77:2 (2013), 3–34; Izv. Math., 77:2 (2013), 223–252
V. V. Volchkov, Vit. V. Volchkov, “Behaviour at infinity of solutions of twisted convolution equations”, Izv. RAN. Ser. Mat., 76:1 (2012), 85–100; Izv. Math., 76:1 (2012), 79–93
V. V. Volchkov, Vit. V. Volchkov, “On the extension problem for solutions of homogeneous convolution equations”, Izv. RAN. Ser. Mat., 75:3 (2011), 65–96; Izv. Math., 75:3 (2011), 507–537
V. V. Volchkov, Vit. V. Volchkov, “On a problem of Berenstein–Gay and its generalizations”, Izv. RAN. Ser. Mat., 74:4 (2010), 33–62; Izv. Math., 74:4 (2010), 691–721
V. V. Volchkov, Vit. V. Volchkov, “Sets with the Pompeiu Property on the Plane and on the Sphere”, Mat. Zametki, 87:1 (2010), 69–82; Math. Notes, 87:1 (2010), 59–70
V. V. Volchkov, Vit. V. Volchkov, “Extremal problems related to the John uniqueness theorem”, Algebra i Analiz, 21:5 (2009), 37–69; St. Petersburg Math. J., 21:5 (2010), 705–729
V. V. Volchkov, Vit. V. Volchkov, “Convolution equations in many-dimensional domains and on the Heisenberg reduced group”, Mat. Sb., 199:8 (2008), 29–60; Sb. Math., 199:8 (2008), 1139–1168
V. V. Volchkov, “Uniqueness theorems for solutions of the convolution equation
on symmetric spaces”, Izv. RAN. Ser. Mat., 70:6 (2006), 3–18; Izv. Math., 70:6 (2006), 1077–1092
V. V. Volchkov, “A local two-radii theorem for quasianalytic classes of functions”, Mat. Zametki, 80:4 (2006), 490–500; Math. Notes, 80:4 (2006), 468–477
2001
31.
V. V. Volchkov, “A definitive version of the local two-radii theorem on hyperbolic spaces”, Izv. RAN. Ser. Mat., 65:2 (2001), 3–26; Izv. Math., 65:2 (2001), 207–229
V. V. Volchkov, “Theorems on the injectivity of the Radon transform on spheres”, Dokl. Akad. Nauk, 354:3 (1997), 298–300
38.
V. V. Volchkov, “Uniqueness theorems for some classes of functions with zero spherical means”, Mat. Zametki, 62:1 (1997), 59–65; Math. Notes, 62:1 (1997), 50–55
V. V. Volchkov, “The final version of the mean value theorem for harmonic functions”, Mat. Zametki, 59:3 (1996), 351–358; Math. Notes, 59:3 (1996), 247–252
V. V. Volchkov, “Approximation of functions on bounded domains in $R^n$ by linear combinations of shifts”, Dokl. Akad. Nauk, 334:5 (1994), 560–561; Dokl. Math., 49:1 (1994), 160–162
V. V. Volchkov, “Theorems on two radii on bounded domains of Euclidean spaces”, Differ. Uravn., 30:10 (1994), 1719–1724; Differ. Equ., 30:10 (1994), 1587–1592
V. V. Volchkov, “New two-radii theorems in the theory of harmonic functions”, Izv. RAN. Ser. Mat., 58:1 (1994), 182–194; Russian Acad. Sci. Izv. Math., 44:1 (1995), 181–192
V. V. Volchkov, “New theorems on the mean for solutions of the Helmholtz equation”, Mat. Sb., 184:7 (1993), 71–78; Russian Acad. Sci. Sb. Math., 79:2 (1994), 281–286
V. V. Volchkov, A. V. Ivanov, N. I. Kislyakov, A. K. Rebrov, R. G. Sharafutdinov, V. A. Sukhnev, “Low-density jets beyond a sonic nozzle at large pressure drops”, Prikl. Mekh. Tekh. Fiz., 14:2 (1973), 64–73; J. Appl. Mech. Tech. Phys., 14:2 (1973), 200–207
V. V. Volchkov, Vit. V. Volchkov, N. P. Volchkova, “On some properties of functions”, Math. Ed., 2022, no. 1(101), 38–47
2021
56.
V. V. Volchkov, Vit. V. Volchkov, N. P. Volchkova, “On some properties of functions characterized by zero integrals”, Math. Ed., 2021, no. 4(100)-2, 38–48