**Sergei Bernstein**'s name is often transliterated as Bernshtein and, just occasionally, as Bernshteyn. His father was Natan Osipovich Bernstein (1836-1891), a medical doctor and also an extraordinary professor at the University of Odessa. The family was Jewish, and Natan Bernstein had been an editor of the Odessa magazine *Zion: Organ of Russian Jews* which had only been published for a year around 1861 before being closed down. It is worth noting that the magazine championed emancipation and assimilation of Jews into Russian society. Natan Osipovich held important positions in Odessa being an alderman of the City Council, the Director of the Talmud Torah, the Director of the city hospital, and an honorary justice of the peace. Sergei was brought up in Odessa but his father died on 4 February 1891 just before he was eleven years old. He graduated from high school in 1898. After this, following his mother's wishes, he went with his elder sister to Paris. Bernstein's sister studied biology in Paris and did not return to the Ukraine but worked at the Pasteur Institute. After one year studying mathematics at the Sorbonne, Bernstein decided that he would rather become an engineer and entered the École d'Electrotechnique Supérieure. However, he continued to be interested in mathematics and spent three terms at Göttingen, beginning in the autumn of 1902, where his studies were supervised by David Hilbert.

Bernstein returned to Paris and submitted his doctoral dissertation *Sur la nature analytique des solutions des équations aux dérivées partielles du second ordre* to the Sorbonne in the spring of 1904. Since he could write French considerably better than German, submitting his thesis in France made sense. The thesis begins with the bold words:-

Today all mathematicians and physicists agree that the field of applications for mathematics knows no limits except those of knowledge itself.

Émile Picard, Henri Poincaré and Jacques Hadamard examined this brilliant piece of work and Picard, as chairman of the examiners, wrote the report. The thesis was a fine piece of work solving Hilbert's Nineteenth Problem. This problem, posed by Hilbert at the International Congress of Mathematicians in Paris in 1900, was on analytic solutions of elliptic differential equations and asked for a proof that all solutions of regular analytical variational problems are analytic. Bernstein received his doctorate from the Sorbonne in 1904 and left Paris to attend the International Congress of Mathematicians in Heidelberg later that year. The Congress lasted 8 August to 13 August 1904 but Bernstein remained at Heidelberg until the spring of 1905 when he went to St Petersburg. His thesis was published in 1904 and in the following year the papers *Sur l'interpolation* and *Sur la déformation des surfaces* were published.

Despite this excellent work, and the fact that he had already received his doctorate, when Bernstein returned to Russia in 1905 he had to start his doctoral programme again since Russia did not recognise foreign qualifications for university posts. In 1906 he passed his Master's examination at St Petersburg but only with difficulty since Aleksandr Nikolayevich Korkin, who examined him on differential equations, expected him to use classical methods of solution (some sources say that Bernstein only passed the examination at the second attempt). He could not find employment in a university and had to settle for teaching at the recently founded Women's Polytechnic College. He taught there for the year 1907 but, although he was very interested in teaching, he felt he deserved a post in a university where both teaching and research were valued.

Some of Bernstein's opinions on teaching are at THIS LINK

He moved to Kharkov in 1908 where he submitted a thesis *Investigation and Solution of Elliptic Partial Differential Equations of Second Degree* for yet another Master's degree. As well as describing his approach to solving Hilbert's 19^{th} Problem, it also solved Hilbert's 20^{th} Problem on the analytic solution of Dirichlet's problem for a wide class of non-linear elliptic equations. Dmitrii Matveevich Sintsov and Antoni-Bonifatsi Pavlovich Psheborski examined his thesis and from that time Bernstein was able to lecture at Kharkov University as a dozent.

However, life was not what Bernstein hoped for, and he wrote to Hilbert saying that his situation was "hopeless". Whether Hilbert sought to help is unclear, but Bernstein received an offer of a position at Harvard University from William Osgood. In the spring of 1910 Bernstein went to Göttingen to talk to Dunham Jackson who was visiting Göttingen from Harvard. It is unclear what happened but Bernstein gave up the chance of going to Harvard and, after his Göttingen visit, returned to Kharkov. In 1913 he received his second doctorate, this time from Kharkov University for his thesis *About the Best Approximation of Continuous Functions by Polynomials of Given Degree*. The thesis had been completed a year earlier and the results of the thesis had earned Bernstein a prize from the Belgium Academy of Science in 1911. This had come about in the following way. Charles-Jean de La Vallée Poussin had asked in 1908: is it possible to approximate the ordinate of a polygonal line by means of a polynomial of degree *n* with error less than ^{1}/_{n}? Both de La Vallée Poussin and Bernstein made some progress in the following years and then the Belgium Academy of Science offered a prize for a solution. Bernstein gave a complete solution in 1911, introducing what are now called the Bernstein polynomials and giving a constructive proof of Weierstrass's theorem (1885) that a continuous function on a finite subinterval of the real line can be uniformly approximated as closely as we wish by a polynomial. He sent his proof to the Belgium Academy of Science and was awarded the prize. He wrote:-

The example of the problem of the best approximation of the function|x|, posed by de la Vallée-Poussin reaffirms, once again, the fact that a well-posed specific question leads to theories of a much more general significance.

He defended his doctoral thesis on 19 May 1913 before a committee chaired by Psheborski. Here is an extract from the speech he made at the defence of the thesis [7]:-

Mathematicians for a long time have confined themselves to the finite or algebraic integration of differential equations, but after the solution of many interesting problems the equations that can be solved by these methods have to all intents and purposes been exhausted, and one must either give up all further progress or abandon the formal point of view and start on a new analytic path. The analytic trend in the theory of differential equations has only recently become established; and only seven years ago the late Professor Korkin in a conversation with me spoke scornfully of the "decadence" of Poincaré's work.

He taught at Kharkov University for 25 years beginning in 1907. He also gave lectures at the Women's College until 1918 (when women were admitted to universities) and, from 1912 to 1918, he also lectured at Kharkov Commercial University. He was made an ordinary professor at Kharkov University in 1920. Around this time he regularly went from Kharkov to Ekaterinoslav to lecture at the 'new' Polytechnic Institute there. In 1924 he was elected as a corresponding member of the Russian Academy of Sciences and, in the following year, became an ordinary member of the Ukrainian Academy of Sciences. In 1927 Gösta Mittag-Leffler died and Bernstein was elected as a corresponding member of the Académie des Sciences in Paris to replace him. He became Director of the Kharkov Mathematical institute in 1928. The Russian Academy of Sciences had been renamed the USSR Academy of Sciences in 1925 and, in 1929, Bernstein - already a corresponding member since 1924 - was elected a full member. Beginning in 1930 the government made academic teaching and research into a political issue and Bernstein made his opposition known in a published response. Almost immediately he was removed as Director of the Kharkov Mathematical institute but, although he was strongly attacked for his views, he continued to hold his chair until 1932. Beginning in 1933 many lecturers and students at Kharkov were arrested and some were shot but, fortunately for Bernstein, he had left just prior to these events.

In 1932 Bernstein left Kharkov to become head of the Department of Probability Theory and Mathematical Statistics of the Mathematical Institute of the USSR Academy of Sciences in Leningrad. He also lectured at Leningrad University from 1934. At the beginning of 1939 he took up a lecturing post at Moscow University but continued to live in Leningrad. He lost his position as Head of the Mathematical Institute in Leningrad but, in 1940, became an honorary member of the Moscow Mathematical Society. However in June 1941 Hitler attacked Russia with the German armies moving rapidly towards Leningrad. Bernstein left Leningrad and went to the town of Borovoe between Astana and Kokshetau in the north of Kazakhstan. His son, German Sergeevich, did not leave the family home in Leningrad and was trapped by the siege of the city which began in September 1941. German Sergeevich died in an attempt to flee from the besieged city and, following the death of his son, Bernstein decided that he would give up the family home in Leningrad and move his residence to Moscow. Bernstein began teaching at the University of Moscow and over the next seven years he worked on editing Pafnuty Chebyshev's complete works. However, in 1947 he was dismissed from the University and became Head of the Department of Constructive Function Theory at the Steklov Institute. He retired in 1957.

At the International Congress of Mathematicians at Cambridge in 1912, Bernstein talked about his work on constructive function theory, which today is called approximation theory. He also talked about 'constructive function theory' in a lecture to the Academy in 1945:-

As constructive function theory we want to call the direction of function theory which follows the aim to give the simplest and most pleasant basis for the quantitative investigation and calculation both of empirical and of all other functions occurring as solutions of naturally posed problems of mathematical analysis(for instance, as solutions of differential or functional equations). In its spirit this direction is very near to the mathematical work of Chebyshev; therefore no wonder that modern constructive function theory uses and develops the ideas of our deceased famous member.

Bernstein continued to develop these ideas, solving problems in interpolation theory, giving methods of mechanical integration and, in 1914, introduced a new class of quasi-analytic functions. Heinrich Begehr writes:-

Some of his main contributions are about the best approximation of continuous functions by polynomials of prescribed degrees, an example of a continuous function the trigonometric interpolation sequence of which is divergent, an estimation of a certain weighted maximum of the derivative of a polynomial on the segment[-1,1]by the one of the polynomial itself, the Bernstein polynomials which turn out not to be interpolation polynomials for the approximated continuous function itself but for a certain smoothed-out function. His later research in this area is devoted to weight functions in connection with the approximation through entire transcendental functions of exponential type.

Some of Bernstein's most important work was in the theory of probability and he wrote an important text *Probability theory* (1911) (4^{th} edition 1946) even before the award of his Russian doctorate. He attempted an axiomatisation of probability theory in 1917 and in 1922-4 gave lectures on probability theory at the Sorbonne. This course of lectures was written up as the book* Leçons sur les propriétés extrémale et la meilleure approximation des fonctions analytique d'une variable réele* (1926). For this brilliant book, Bernstein was awarded a prize by the Académie des Sciences in Paris. During the two years 1922-24, as well as visiting Paris, he visited Germany. As to his many other contributions to probability theory, he generalised Lyapunov's conditions for the central limit theorem, studied generalisations of the law of large numbers, and worked on Markov processes and stochastic processes. Yuri Linnik gives the following summary of Bernstein's contributions to probability in [18]:-

The theory of probability is indebted to S N Bernstein for fundamental contributions on a number of topics; the axiomatic theory of probability, the foundations of normal correlation using limit theorems and the development of the general theory of correlation, the extension of the central limit theorem to sums of stochastically dependent variables, especially to heterogeneous Markov chains, and stochastic differential equations; the application of the theory of probability to biology and economics and applications of the methods of the theory of probability to the constructive theory of functions.

N D Kazarinoff writes about Linnik's overview of of probability theory [17]:-

[

Linnik]notes that Bernstein's paper "An attempt at axiomatizing the foundations of the theory of probability"(1917)is perhaps the first paper directly on this subject. The approach is from the point of view of algebraic structure.

Bernstein also studied applications of probability, in particular to genetics. An important paper he wrote on this topic is *Mathematical problems in modern biology *(1922) which contains what became known as the Bernstein problem. He proved a special case of his own problem in *Solution of a mathematical problem related to the theory of inheritance* (1924).

Bernstein's *Collected Works* appeared as a four-volume work between 1952 and 1964. The volumes were edited by Bernstein himself [25]:-

From1950all Bernshtein's scientific activities have been bound up with the preparation of his works for publication. It is difficult even to imagine how much intricate work he has done in editing and annotating his papers. Many of the annotations differ considerably from the usual kind of footnote; they contain a series of valuable ideas and observations and are to be regarded as new scientific work.

We have mentioned above a number of honours awarded to Berstein but we now list a few others. He was awarded an honorary doctorate by Algiers University in 1944, and one by the Sorbonne in the following year. He was elected a full member of the Académie des Sciences in Paris in 1955. In 1942 he was given a 'State Award: First Class' and in the same year received the Stalin Prize for his three papers: *On the sums of dependent variables with almost zero correlation*; *On the approximation of continuous functions by the linear differential operator from a polynomial*; and *On Fisher's provable probabilities*. In addition he received two Orders of Lenin and an Order of the Red Banner of Labour. As a final thought, one has to ponder the way that Bernstein was both honoured by the government and also badly treated by the government because he did not fall into line with the official political line.

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

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