**Dmitrii Evgenevich Menshov**'s father, Evgenii Titovich Menshov, was a medical doctor who worked in the New Ekaterininskii Hospital and the Lazarevskii Institute of Oriental Languages. His mother, Alexandra Nikolaevna Tatishcheva, was an important early influence in Menshov's education. She was a highly educated woman who, as well as educating her son, also sometimes gave French lessons to other children.

In 1904, at the age of 12 years, Menshov began his secondary schooling. He attended the gymnasium section of the Lazarevskii Institute of Oriental Languages where his father acted as school doctor. Influenced by his mother's tuition in foreign languages, Menshov's first love at school was indeed for languages. He went on to study French, German, English, Latin, and Armenian at school. However, Menshov had an outstanding mathematics teacher and, like many children who are influenced by an outstanding teacher, Menshov began to show a strong interest in mathematics from about the age of 13. He also was strongly attracted to physics so, with wide interests across many subjects, he graduated from the school in 1911 with the gold medal for outstanding achievement.

After leaving school, Menshov sat the entrance examination for the Moscow Engineering College and began his studies there in the autumn of 1911. However he only studied there for six months before deciding to leave and work on his own on learning advanced mathematics. Then in the autumn of 1912 Menshov entered the Department of Physics and Mathematics at Moscow University. There he attended lectures by Egorov, Lakhtin, Andreev and he took his first course on functions of a real variable given by Byushgens. Perhaps the most significant event for Menshov, however, was that Luzin returned from Göttingen to Moscow in the autumn of 1914 and began to lecture on functions of a real variable.

Menshov attended Luzin's lecture course, and when Luzin posed the open problem of whether the Denjoy integral and the Borel integral were equivalent, he was able to solve the problem. The Denjoy integral is the more general of the two and Menshov showed that this was the case. He showed Luzin his solution to the problem that Luzin had just posed and before the end of 1914 the two had begun a firm mathematical friendship. Menshov's discovery, made while still an undergraduate, became his first publication. It appeared as the paper *The relationship between the definitions of the Denjoy and Borel integrals* in 1916.

Luzin quickly established a School of Mathematics at Moscow University and Menshov became one of his fist research students along with P S Aleksandrov, M Ya Suslin, and A Ya Khinchin. Menshov's first degree was awarded in 1916 for the thesis which he wrote on *The Riemann theory of trigonometric series* which was examined by Egorov and Luzin. However, only three weeks after he graduated, Menshov discovered one of his most fundamental results on the uniqueness problem for trigonometric series. Let us describe this result.

Consider the trigonometric series

*a*

_{0}/2 + ∑ (

*a*

_{n}cos

*nx*+

*b*

_{n}sin

*nx*).

*x*in [0, 2π] -

*E*, for a countable set

*E*, then

*a*

_{n}=

*b*

_{n}= 0 for all

*n*. Vallée Poussin had proved that if the above series converged to a finite Lebesgue integrable function

*f*(

*x*) then the given series is the Fourier series of

*f*(

*x*). It was expected that Vallée Poussin's result would still hold if the countable set

*E*was replaced by a set

*E*of measure zero. The remarkable, and unexpected, result that Menshov discovered in 1916 was that this was not so, for he constructed a trigonometric series which converges to 0 for all

*x*in [0, 2π] -

*E*, for a set

*E*of measure zero, yet not all the coefficients of the trigonometric series are zero.

By the end of 1918 Menshov had been awarded his Master's degree and he went to Ivanovo north-east of Moscow, which at that time was the temporary capital of the revolutionary government, but he soon moved to Nizhnii-Novgorod where he was appointed as a professor at the University. He taught at Nizhnii-Novgorod during 1919 and early 1920 but he returned to Ivanovo in May 1920 where he was appointed as a professor at the Ivanovo Pedagogic Institute. In addition to this appointment he also taught at the Polytechnic Institute at Ivanovo from January 1921. At this time Luzin and other members of his research school were in Ivanovo so Menshov was certainly in the mainstream of the exciting mathematics that was being developed.

In the autumn of 1922 Menshov returned to Moscow and began teaching at the University. He also taught for a few years at the Moscow Institute of Forest Technology. It may have been noticed by an attentive reader that we have still not noted that Menshov being awarded a doctorate (equivalent to the habilitation or D.Sc.). In fact he never submitted a thesis for a doctorate but, despite this, he was awarded the doctorate in 1935 since ([1] or [2]):-

Together with the award of the doctorate came Menshov's appointment to a professorship at Moscow University.... he had already been acknowledged as one of the world's most outstanding specialists on the theory of functions of a real and a complex variable.

In 1933 a new chair of Analysis and Theory of Functions was created at Moscow University and Lavrent'ev appointed. In 1938 the Faculty of Mechanics and Mathematics at Moscow University founded two chairs, the chair of the Theory of Functions and the chair of Functional Analysis. Privalov held this first chair up to 1941 but then, on Privalov's early death in that year, Menshov was appointed to the chair of the Theory of Functions. Lusternik held the chair of Functional Analysis from 1938. In 1943 these two chairs were combined and the Department of Theory of Functions and Functional Analysis was created with Menshov as its head. Menshov also worked a the Steklov Mathematical Institute of the USSR Academy of Sciences from 1934 to 1941 and then again from 1947.

Menshov's mathematical interests and the style of his mathematics is described in ([3] and [4]):-

For his work on the representation of functions by trigonometric series, Menshov was awarded a State Prize in 1951. He was then elected a Corresponding Member of the USSR Academy of Sciences in 1953. In 1958 Menshov attended the International Congress of Mathematicians in Edinburgh and he was invited to address the Congress with his paperHis scientific interests relate principally to the theory of trigonometric series, the theory of orthogonal series and the problem of monogenity of functions of a complex variable. He published more than eighty papers on these subjects, which have had an exceptionally great effect on the development of the whole theory of functions. Menshov does not belong among the ranks of those mathematicians who undertake the solution of comparatively easy problems, or who continue the research of other authors on a course that has already been indicated. A characteristic feature of scientific activity is that in his work on the theory of functions he solved a number of extremely difficult key problems which had baffled many eminent mathematicians.

*On the convergence of trigonometric series*.

The first of the two pictures of Menshov which we have given was taken while he was at the Congress in Edinburgh, Scotland in 1958.

**Article by:** *J J O'Connor* and *E F Robertson*

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