M. I. Dyachenko, B. S. Kashin, T. P. Lukashenko, M. G. Plotnikov, A. P. Solodov, N. N. Kholshchevnikova, “Valentin Anatol'evich Skvortsov (on his ninetieth birthday)”, Russian Math. Surveys, 80:6 (2025), 1131

On 25 June 2025 the renowned mathematician, Doctor of Sciences (physical and mathematical sciences), distinguished professor of Lomonosov Moscow State University Valentin Anatol’evich Skvortsov observed his 90th birthday. He made a considerable contribution to the theory of measure and integrals, the theory of orthogonal series and harmonic analysis. His results were decisive in the solution of a number of crucial questions in generalized integration and in the field of representation of functions by series in general and special orthogonal systems. Investications performed by Skvortsov and his school develop further, on the contemporary basis, the line in real analysis going back to classical works by E. Borel, Denjoy, Luzin, Kolmogorov, and Menshov.

He was born in a small town of Volosovo, Leningrad Oblast’, in teachers’ family. He graduated from the local secondary school with a gold medal of distinction and in 1953 enrolled in the Faculty of Mechanics and Mathematics at Moscow State University. Already during his student years he turned to research work, with Dmirii Evgen’evich Menshov as his scientific advisor. Soon he obtained results on generalized integration that immediately attracted attention of experts.

Various generalizations of the Lebesgue integral appeared in the first half of the 20th century, and the two main problems in analysis that were to be solved with their use were the reconstruction of a function from its derivative and the recovery of the coefficients of a trigonometric series from its sum by means of generalized Fourier formulae. In his first paper “Interrelation between Denjoy’s general integral and totalization $(T_{2s})_0$” Skvortsov proved (which was an unexpected result at the time) that so-called ‘trigonometric integrals’ which solve the second problem (for example, Denjoy’s totalization $(T_{2s})$) do not agree with the wide Denjoy integral, which is connected with the first problem: a function can be integrable in both senses, but the values of integrals can be distinct. This paper presented results obtained in Skvortsov’s diploma thesis and was submitted by A. N. Kolmogorov for publishing in the journal Doklady Akademii Nauk in 1959.

Subsequently, Skvortsov and his students (T. P. Lukashenko, V. A. Sklyarenko, and some others) showed that many results in the theory of trigonometric series valid for the Lebesgue integral also hold for the narrow Denjoy integral, but cease to be valid for the wide Denjoy integral.

In 1964 Skvortsov obtained his first result on the representation of functions by orthogonal series: he proved that if a series in the Haar system converges to zero everywhere, then all of its coefficients are equal to zero. This result, an analogue of Cantor’s uniqueness theorem for the Haar system, was claimed by Haar himself in 1910, but his original proof contained an error. This was noticed by participants of the seminar on the theory of functions at the Faculty of Mechanics and Mathematics at Moscow State University, and soon four papers presenting a valid proof, including one by Skvortsov, were published at approximately the same time.

Skvortsov’s further works in this direction made a decisive contribution to the solution of problems related to the uniqueness of representations of functions by series in the Haar and Walsh systems (which, along with the trigonometric system, play a major part in harmonic analysis), as well as by series in Vilenkin system and systems of characters of zero-dimensional compact groups. A characteristic feature of these works is the infiltration of ideas and methods from the theories of generalized integration and differentiation to the theory of orthogonal series. One of these methods, developed by Skvortsov himself and producing some interesting results, reduces the analysis of the convergence of orthogonal series to some questions on differentiation of certain related set functions.

A well-known result of Skvortsov’s in the uniqueness theory for orthogonal series is his construction of the HD-integral, which solves the problem of the recovery of an everywhere convergent Haar or Walsh series from its sum. Subsequently, by developing methods of the construction of integrals in terms of generalized Riemann sums due to Kolmogorov, J. Kurzweil, and R. Henstock, he developed a unified approach to the reconstruction of an orthogonal series from its sum for a wide class of systems. Another important result of Skvortsov was his construction of a perfect $M$-set of measure zero for the Walsh system, and at the same time of a null series in this system. In the 1970s he obtained the first uniqueness results for the Haar and Walsh systems in the multivariate case. To date, not much has been achieved in this direction for multidimensional systems of functions (nor even for the multiple trigonometric system), and many fundamental questions are still open. By generalizing expectation in the sense of the $A$-integral, introduced by Kolmogorov, to the case of condition expectation, Skvortsov investigated martingales with respect to the $A$-integral.

A number of important results on Haar and Walsh series are due to Skvortsov’s students M. G. Plotnikov, N. A. Bokaev, T. Sworowska, V. V. Koston, I. V. Polyakov, and others.

In the 1980s–2000s Skvortsov turned to the general theory of integrals. In conjunction with his coauthors B. Bongiorno and L. Di Piazza and with his students F. Tulone, P. Sworowski, and Yu. A. Zhereb’ev, he obtained new descriptive characterizations of the Lebesgue, Denjoy–Perron, and some other integrals in terms of the absolute continuity of the so-called variational measure. With A. Boccuto they constructed Henstock-type integrals for functions taking values in vector lattices. In the Banach-valued case Skvortsov, his student A. P. Solodov, and also K. M. Naralenkov examined the dependence of the properties of such integrals on the structure of the target space. The properties of Henstock integrals in infinite-dimensional spaces were the subject of a paper by Skvortsov amd P. Maldoni, and one of Skvortsov’s results, established in collaboration with Henstock and Maldoni, completed the construction of the Henstock integral in the space $R^{[0,1]}$.

In the 2000s Skvortsov obtained a number of results (some of them with Tulone) on the representation of functions by series in the systems of characters of zero-dimensional compact groups. Not so long ago, in a joint work with N. N. Kholshchevnikova they solved the problem of categories for such systems.

As Skvortsov himself notes, a result particularly special to him is the complicated and delicate construction of a non-trivial Walsh series with coefficients tending to zero such that a certain subsequence of its partial sums converges to zero at each point. The question of whether or not there exists a trigonometric series with similar properties was posed by many famous experts, beginning with P. L. Ul’yanov, but only in 2020 did G. Kozma and A. M. Olevskii construct an example.

Skvortsov is the author of approximately 200 research papers, including the well-known monograph Walsh series and transforms. Theory and application, written in collaboration with B. I. Golubov and A. V. Efimov, and the monograph Generalized integrals written with his former students Solodov and Lukashenko. He was an invited speaker at many international conferences on real analysis and harmonic analysis. He is a member of the editorial boards of the journals European Journal of Mathematics, Eurasian Mathematical Journal, and Fundamental’naya i Prikladnaya Mathematica.¹ Skvortsov’s research in the field of real analysis was recognized in a yearly award of the editorial board of the journal Real Analysis Exchange. As a member of the Department of the Theory of Functions and Functional Analysis of Lomonosov Moscow State University, Skvortsov was a scientific advisor of 20 Ph.D.’s, and five of his former students defended their D.Sc. theses.

Skvortsov has made a significant contribution to the development of school mathematical education and to the olympiad movement. In the 1990s he was a member of the editorial board of the journal Matematika v Shkole (Mathematics in Schools). He was active in the organization of of a boarding school under the auspices of the Moscow University, founded by Kolmogorov in 1963, and he taught there for several years. For many years he was engaged in the organization of All-Union and international mathematical olympiads for schoolchildren. And in the period of 1971–1975 he was the head of the Soviet team at international mathematical olympiads. Together with E. A. Morozova and I. S. Petrakov, he published the book International mathematical olympiads. Another publication addressed to high-school students was his pamphlet Examples of metric spaces published (in Russian) in 2002. Skvortsov was actively involved in the creation of a number of English textbooks for mathematics students, which are in use in the Faculty of Mechanics and Mathematics at Moscow State University.

Skvortsov’s interests extend far away from mathematics. In his youth he was among the leading actors of the student theater on Lenin Hills, and in 1969, on his initiative and with his active engagement the Club of Scientists of the Moscow State University was organized, where he is still a co-chairmen.

We greet Valentin Abatol’evich Skvortsov heartily on his 90th birthday and wish him health, prosperity, and long creative life!

Citation: M. I. Dyachenko, B. S. Kashin, T. P. Lukashenko, M. G. Plotnikov, A. P. Solodov, N. N. Kholshchevnikova, “Valentin Anatol'evich Skvortsov (on his ninetieth birthday)”, Russian Math. Surveys, 80:6 (2025), 1131–1134

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