Russian Mathematical Surveys, 2025, Volume 80, Issue 2, Pages 359–364 DOI: https://doi.org/10.4213/rm10221e (Mi rm10221)

DOI: https://doi.org/10.4213/rm10221e

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On 25 September 2024 doctor of physical and mathematical sciences, Professor Valerii Vladimirovich Volchkov observed his 60th birthday.

Virtually all of his life has been connected with Donbas. He spent his early childhood in Donetsk and his school years in Shakhtersk (Donetsk Oblast’). Then he moved to Donetsk again with his family. Already during his early teenage years he showed an immense desire for creative work: he assembled a wood lathe on his own, on which he carved figures of complicated shape, conducted chemistry experiments at home, built a radio receiver, was fond of belletristic, and graduated from music school. However, he only found his true calling – doing mathematics – in eighth grade, when he read the books Geometry revisited by H. S. M. Coxeter and S. L. Greitzer and Invitation to number theory by O. Ore. He obtained his first mathematical results in 1980–1981 when, as a 16-years old high-school student he rediscovered quite a number of known geometric results and proved more than 50 new theorems on special lines and points in geometry. These results revealed his potentials and were subsequently used in various courses for high-school and university students and school teachers.

In 1981 Volchkov enrolled in the Faculty of Mathematics at Donetsk State University. From his first days there he enjoyed respect of students and professors. On an invitation of associate professor A. K. Slipenko he became the leader of a student study group on number theory, in parallel with his own research and his work at professor A. Ya. Savchenko’s seminar on stability theory at the Institute of Applied Mathematics and Mechanics. As a third-year student he decided to major in approximation of function, with R. M. Trigub as his scientific advisor and began attending trigob’s research seminar. However, when B. D. Kotlyar from Dnepropetrovsk made a report at this seminar, Volchkov’s interests turned to integral geometry and its applications. During he fourth and fifth years at the university he obtained results on geometric criteria for holomorphy, subsequently published in the journal Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika.^1[x]1Translated into English in Journal of Mathematical Sciences. New York.

In 1986 Volchkov graduated from Donetsk State University, with specialization of a mathematician. Subsequently, recalling his school and university years he said that his education was of a very special nature and joked: “I did not solve textbook problems, but was always busy failing to solve well-known mathematical problems.”

In the fall of 1986 Volchkov was conscripted and served in the Soviet army group in Easy Germany until 1988. Far from home, isolated, and without access to mathematics literature, he continued his research. At that period of time he managed to proved the following result on functions with vanishing integrals over all closed unit cubes lying in an open ball of radius $r$ in $\mathbb{R}^n$, $n\geqslant2$. He showed that for $r\geqslant\sqrt{n+3}/2$ the function must identically be zero, while for $r<\sqrt{n+3}/2$ there exist non-trivial $C^{\infty}$-functions satisfying this condition. In other words the injectivity radius of the Pompeiu transform associated with the indicator of the unit cube in $\mathbb{R}^n$ is $\sqrt{n+3}/2$. This was the historically first explicit value of the injectivity radius, which refined a theorem of C. A. Berenstein and R. Gay from the paper “Le problème de Pompeiu local” (J. Anal. Math., carte blanche to his student. As a result, during his postgraduate years Volchkov laid foundations of his own approaches to a number of problems in integral geometry and the theory of mean periodic functions. He defended his Ph.D. thesis “Pompeiu-type problems in bounded domains” in 1991, before the end of his postgraduate term.

From September 1991 to these days Volchkovs has been working in the Faculty of Mathematics at Donetsk State University. We was awarded the D.Sc. degree in April 1997. In his review of Volchkov’s doctoral thesis “Pompeiu transform and its applications to the theory of functions” L. Zalcman, the president of the Israel Mathematical Society, wrote: “In his works V. V. Volchkov obtained results could not even been imagined by his predecessors. In fact, if I could, I would gave the highest possible mark to this paper.” In 1998 Volchkov was awarded a medal and an international prize of the European Academy for his research.

During the almost 30 years of his further research Volchkov obtained fundamental results on integral geometry, the theory of multidimensional convolution-type equations, and their applications, which have determined to a significant extent the current lines of development of these areas and have brought him high reputation.

Volchkov’s results in integral geometry are related to the recovery of functions from given integral means. This line of research goes back to the first half of the 20th century, to works by G. Minkowski, P. Funk, J. Radon, D. Pompeiu, F. John, L. Zalcman, C. A. Berenstein, and other authors. However, the investigations of similar setups in bounded domains encountered significant problems because the existing methods were based on the Fourier transformation. Volchkov developed new methods, based on expansions in special functions. Using them he managed to solve fully a number of well-known problems (the Zalcman problem [3], [5], the local two-radii problem [4], [6], the support problem [7], [8], the problem of the injectivity of the Radon transform on spheres [3], [4], [9], the description of injectivity sets of the Pompeiu transform [3], [10], Berenstein–Gay problems [8], [11], and some others). These results were highly estimated by well-known experts from many countries. For instance. Zalcman, in his bibliographic survey “Supplementary bibliography to ‘A bibliographic survey of the Pompeiu problem’ ” (Contemp. Math., 278, 2001, 69–74) mentioned Volchkov’s extraordinary contributions and pointed out that the list of Volchkov’s papers took up one third of the whole bibliography on the problem in question. One telling proof of the high importance of Volchkov’s results were the words of E. T. Quinto, a well-known American researcher, that Volchkov’s work compelled him to learn the Russian language.

Volchkov obtained many deep and definitive results on convolution equations on Euclidean and symmetric spaces. Convolution equations are rather important for numerous applications, and such authors as J. Delsart, L. Schwartz, L. Ehrenpreis, B. Malgrange, L. Hörmander, A. F. Leont’ev, C. A. Berenstein, A. Sitaram, and some others published works on this subject. In his monograph Integral geometry and convolution equations [3] (2003) Volchkov improved significantly results due to a number of authors and proposed about 50 new open problems, which immediately attracted the attention of experts. Some of these problems have eventually been solved, which has significantly enriched the theory and indicated new applications. This work was summarised in V. V. Volchkov’s joint monographs with Vit. V. Volchkov Harmonic analysis of mean periodic functions on symmetric spaces and the Heisenberg group [8] and Offbeat integral geometry on symmetric spaces [4]. Among Volchkov’s most important results on convolution equations we can mention uniqueness theorems, results on the structure of the general solution, theorems on the asymptotic behaviour of solutions, removable singularity theorems, and also similar results for systems of convolution equations.

All of these areas do not cover the full scope of Volchkov’s mathematical interests. We discuss briefly some of his result in other areas.

1. Approximation theory. Volchkov obtained $L^p$-analogues of N. Wiener’s classical theorem on the closures of shifts on bounded domains and sharp results on approximation by spherical waves [3], [12], proved some results on sharp constants in $L^2$-approximation [13], and established theorems on approximation of functions by polynomials with integer coefficients [14] and an analogue of T. Carleman’s theorem on tangential approximation [15].

2. Harmonic analysis. Volchkov obtained definitive results on weakly lacunary multiple trigonometric series and their analogues for series in special functions [3], [16]. He also refined a result due to Ehrenpreis and F. Mautner on spectral synthesis on the group of conformal automorphisms of the unit disc [17].

3. Harmonic functions. Volchkov considered some problems concerning ball means in the theory of harmonic functions. His main theorems there were a significant improvement of well-known results due to Delsart (on two radii) and L. Flatto (on one radius) [3], [18].

4. Partial differential equations. Volchkov established new mean value theorems and sharp uniqueness theorems for some equations of mathematical physics (Darboux’s equation, the wave equation, the heat equation, and some others) [3], [4], [19], [20], and solved a problem of Berenstein of the description of the set of solutions to a system of difference-differential equations involving partial derivatives [3].

5. Measure-preserving maps. He found sharp conditions ensuring the measure preservation property for some classes of maps of multidimensional domains [3].

6. Complex analysis. Volchkov established criteria for the solvability of interpolation problems for some classes of functions [4], [21], [22]. We can also mention new Morera-type theorems (including multidimensional ones) and a refinement of a theorem of V. K. Dzyadyk on the geometric description of analytic functions [3], [23].

7. Combinatorial geometry. Volchkov obtained some results on estimates for the density of packings of bounded sets [3].

8. Number theory. He established a new criterion for the validity of the Riemann hypothesis [24], found estiamtes for the irrationality measure of some constants, and proved asymptotically sharp theorems on the mean distribution of the prime numbers.

Note that on pp. 81–90 of the monograph The Riemann hypothesis: A resource for the afficionado and virtuoso alike by P. Borwein, S. Choi, B. Rooney, and A. Weirathmueller (Springer, 2007), the authors presented a time scale of the most important results on the Riemann conjecture, starting with ones due to Euler, Goldbach, Gauss, and so on. One of the achievements marked on this scale is Volchkov’s criterion.

Apart from research work, Volchkov’s scientific, public, and teaching activities are quite multifaceted. For many years he has been a member of the editorial boards of various research journals and of dissertation councils and supervised international research projects. He visited many universities worldwide with lectures and talks on his results. As a brilliant lector, capable of explaining complicated mathematical issues concisely and clearly, he is very popular among the students of Donetsk State University and the staff of Donetsk Science Center. His lectures on number theory, where he presented the theory of the Riemann zeta function, I. M. Vinogradov’s solution of the Goldbach ternary problem, and A. O. Gelfond’s solution of Hilbert’s 7th problem, were particularly memorable.

Volchkov pays great attantion to work with talented youth: he participates in the organization of mathematical olympiads of various levels and, for many years, was the chair of the jury of the Ukrainian contest of the Junior Academy of Sciences. Methods developed by Volchkov are used in the studies of his students and followers.

Volchkov is a Merited Professor of Donetsk State University and an Excellent Worker of Public Education. For his contribution to the development of education and achievements in preparing students to olympiads he was awarded a Honorary Diploma of the Department of Education and Science of Donetsk Oblast’. His talents are combined with modesty, friendliness, and kindness. All of his colleagues note his multifaceted interests, commitment, and unfailing benevolence. Valerii Vladimirovich Volchkov is full of energy, creative ideas, and plans. We wish him new outstanding results, good health, and happiness.

Bibliography
1.	V. V. Volchkov, “Extremal problems on Pompeiu sets. II”, Sb. Math., 191:5 (2000), 619–632
2.	V. V. Volchkov, “On polyhedra with the local Pompeiu property”, Dokl. Math., 62:1 (2000), 69–71
3.	V. V. Volchkov, Integral geometry and convolution equations, Kluwer Acad. Publ., Dordrecht, 2003, xii+454 pp.
4.	V. V. Volchkov and Vit. V. Volchkov, Offbeat integral geometry on symmetric spaces, Birkhäuser/Springer Basel AG, Basel, 2013, x+592 pp.
5.	V. V. Volchkov, “On a problem of Zalcman and its generalizations”, Math. Notes, 53:2 (1993), 134–138
6.	V. V. Volchkov, “Local two-radii theorem in symmetric spaces”, Sb. Math., 198:11 (2007), 1553–1577
7.	V. V. Volchkov, “Solution of the support problem for several function classes”, Sb. Math., 188:9 (1997), 1279–1294
8.	V. V. Volchkov and Vit. V. Volchkov, Harmonic analysis of mean periodic functions on symmetric spaces and the Heisenberg group, Springer Monogr. Math., Springer-Verlag London, Ltd., London, 2009, xii+671 pp.
9.	V. V. Volchkov, “Injectivity sets for the Radon transform over a sphere”, Izv. Math., 63:3 (1999), 481–493
10.	V. V. Volchkov, “Injectivity sets of the Pompeiu transform”, Sb. Math., 190:11 (1999), 1607–1622
11.	V. V. Volchkov and Vit. V. Volchkov, “On a problem of Berenstein–Gay and its generalizations”, Izv. Math., 74:4 (2010), 691–721
12.	V. V. Volchkov, “Approximation of functions on bounded domains in $R^n$ by linear combinations of shifts”, Dokl. Math., 49:1 (1994), 160–162
13.	V. V. Volchkov, “On exact constants in Jackson-type inequalities in the space $L^2$”, Ukrainian Math. J., 47:1 (1995), 125–129
14.	R. M. Trigub and V. V. Volchkov, “Best approximation of constants by polynomials with integer coefficients”, J. Approx. Theory, 2025 (to appear)
15.	V. V. Volchkov and Vit. V. Volchkov, “Approximation of functions on rays in $\mathbb{R}^n$ by solutions to convolution equations”, Siberian Math. J., 64:1 (2023), 48–55
16.	V. V. Volchkov, “Uniqueness theorems for multiple lacunary trigonometric series”, Math. Notes, 51:6 (1992), 550–552
17.	V. V. Volchkov and Vit. V. Volchkov, “Spectral synthesis on the group of conformal automorphisms of the unit disc”, Sb. Math., 209:1 (2018), 1–34
18.	V. V. Volchkov, “New two-radii theorems in the theory of harmonic functions”, Russian Acad. Sci. Izv. Math., 44:1 (1995), 181–192
19.	V. V. Volchkov, “New theorems on the mean for solutions of the Helmholtz equation”, Russian Acad. Sci. Sb. Math., 79:2 (1994), 281–286
20.	V. V. Volchkov and Vit. V. Volchkov, “A uniqueness theorem for the non-Euclidean Darboux equation”, Lobachevskii J. Math., 38:2 (2017), 379–385
21.	V. V. Volchkov and Vit. V. Volchkov, “Spherical means on two-point homogeneous spaces and applications”, Izv. Math., 77:2 (2013), 223–252
22.	V. V. Volchkov and Vit. V. Volchkov, “Interpolation problem with knots on a line for solutions of a multidimensional convolution equation”, Lobachevskii J. Math., 44:8 (2023), 3630–3639
23.	V. Volchkov and Vit. Volchkov, “Zalcman's problem and related two-radii theorems”, Anal. Math. Phys., 13:5 (2023), 72, 47 pp.
24.	V. V. Volchkov, “On an equality equivalent to the Riemann hypothesis”, Ukrainian Math. J., 47:3 (1995), 491–493

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\paper Valerii Vladimirovich Volchkov (on his sixtieth birthday)
\jour Russian Math. Surveys
\yr 2025
\vol 80
\issue 2
\pages 359--364
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