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Sergi Bernstein graduated from high school in 1898. Following this, he went to Paris where he studied at the Sorbonne and at Ecole d'Electrotechnique Supérieure. During his work in Paris, Bernstein spent the session 19021903 at Göttingen.
Bernstein's doctoral dissertation submitted to the Sorbonne was a fine piece of work solving Hilbert's Nineteenth Problem. This problem, posed at the 1900 Congress, was on analytic solutions of elliptic differential equations. He received his doctorate from the Sorbonne in 1904.
Despite this excellent piece of work in his doctoral thesis, 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. He studied for his Master's degree at Kharkov, continuing his way through Hilbert's Problems by solving the Twentieth on the analytic solution of Dirichlet's problem for a wide class of nonlinear elliptic equations.
In 1918 Bernstein was awarded his Master's degree and then, in 1913, he received his second doctorate, this time from Kharkov. He taught at Kharkov University for 25 years beginning in 1907.
From 1933 he lectured at Leningrad University and also lectured at the Polytechnic Institute. During this time he worked at the Mathematical Institute of the U.S.S.R. Academy of Sciences. In 1943 Bernstein moved to the University of Moscow and over the next seven years he worked on editing Chebyshev's complete works.
Bernstein worked on the theory of best approximation of functions. He greatly extended work begun by Chebyshev in 1854. In 1911 he introduced what are now called the Bernstein polynomials to give a constructive proof of Weierstrass' theorem (1885), namely that a continuous function on a finite subinterval of the real line can be uniformly approximated as closely as we wish by a polynomial.
At the International Congress at Cambridge in 1912, Bernstein talked about this work. He then continued to develop these ideas, solving problems in interpolation theory, methods of mechanical integration and, in 1914, introduced a new class of quasianalytic functions.
Some of Bernstein's most important work was in the theory of probability. He attempted an axiomatisation of probability theory in 1917.
He generalised Lyapunov's conditions for the central limit theorem, studied generalisations of the law of large numbers, worked on Markov processes and stochastic processes.
Bernstein also studied applications of probability, in particular to genetics.
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