Sergei Natanovich Bernstein

Sergei Natanovich Bernstein (Серге́й Ната́нович Бернште́йн, sometimes Romanized as Bernshtein; 5 March 1880 – 26 October 1968) was a Russian and Soviet mathematician of Jewish origin known for contributions to partial differential equations, differential geometry, probability theory, and approximation theory.

Partial differential equations
In his doctoral dissertation, submitted in 1904 to the Sorbonne, Bernstein solved Hilbert's nineteenth problem on the analytic solution of elliptic differential equations. His later work was devoted to Dirichlet's boundary problem for non-linear equations of elliptic type, where, in particular, he introduced a priori estimates.

Probability theory
In 1917, Bernstein suggested the first axiomatic foundation of probability theory, based on the underlying algebraic structure. It was later superseded by the measure-theoretic approach of Kolmogorov.

In the 1920s, he introduced a method for proving limit theorems for sums of dependent random variables.

Approximation theory
Through his application of Bernstein polynomials, he laid the foundations of constructive function theory, a field studying the connection between smoothness properties of a function and its approximations by polynomials. In particular, he proved the Weierstrass approximation theorem and Bernstein's theorem (approximation theory).

Publications

 * S. N. Bernstein, Collected Works (Russian):
 * vol. 1, The Constructive Theory of Functions (1905–1930), translated: Atomic Energy Commission, Springfield, Va, 1958
 * vol. 2, The Constructive Theory of Functions (1931–1953)
 * vol. 3, Differential equations, calculus of variations and geometry (1903–1947)
 * vol. 4, Theory of Probability. Mathematical statistics (1911–1946)
 * S. N. Bernstein, The Theory of Probabilities (Russian), Moscow, Leningrad, 1946