Sergei Natanovich Bernstein | |
---|---|

Sergei Natanovich Bernstein Sergei Natanovich Bernstein | |

Born |
Odessa, Kherson Governorate, Russian Empire | 5 March 1880

Died |
26 October 1968 Moscow, Soviet Union | (aged 88)

Residence | Russian Empire, Soviet Union |

Nationality | Soviet |

Alma mater | University of Paris |

Known for |
Bernstein's inequality in analysis Bernstein inequalities in probability theory Bernstein polynomial Bernstein's theorem (approximation theory) Bernstein's theorem on monotone functions Bernstein problem in mathematical genetics |

**Sergei Natanovich Bernstein** (Russian: 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.^{[1]}^{[2]}

## Work

### 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.^{[3]} 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the first axiomatic foundation of probability theory, based on the underlying algebraic structure.^{[4]} It was later superseded by the measure-theoretic approach of Kolmogorov.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.^{[5]} In particular, he proved
the Weierstrass approximation theorem^{[6]}^{[7]} 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)

- vol. 1,
- S. N. Bernstein,
*The Theory of Probabilities*(Russian), Moscow, Leningrad, 1946

## See also

- A priori estimate
- Bernstein algebra
- Bernstein's inequality (mathematical analysis)
- Bernstein inequalities in probability theory
- Bernstein polynomial
- Bernstein's problem
- Bernstein's theorem (approximation theory)
- Bernstein's theorem on monotone functions
- Bernstein–von Mises theorem
- Stone–Weierstrass theorem

## Notes

- ↑ Youschkevitch, A. P.. "BERNSTEIN, SERGEY NATANOVICH".
*Dictionary of Scientific Biography*. http://www.encyclopedia.com/doc/1G2-2830904824.html. - ↑ Lozinskii, S. M. (1983). "On the hundredth anniversary of the birth of S. N. Bernstein". p. 163. Digital object identifier:10.1070/RM1983v038n03ABEH003497.
- ↑ Akhiezer, N.I.; Petrovskii, I.G. (1961). "S. N. Bernshtein's contribution to the theory of partial differential equations". http://iopscience.iop.org/0036-0279/16/2/A01.
- ↑ Linnik, Ju. V. (1961). "The contribution of S. N. Bernšteĭn to the theory of probability". pp. 21–22. Digital object identifier:10.1070/rm1961v016n02abeh004103. MR0130818.
- ↑ Videnskii, V. S. (1961). "Sergei Natanovich Bernshtein — founder of the constructive theory of functions". p. 17. Digital object identifier:10.1070/RM1961v016n02ABEH004102.
- ↑ S. Bernstein (1912–13) "Démonstration du théroème de Weierstrass, fondeé sur le calcul des probabilités,
*Commun. Soc. Math. Kharkow*(2) 13: 1-2 - ↑ Kenneth M. Lavasseur (1984) A Probabilistic Proof of the Weierstrass Theorem, American Mathematical Monthly 91(4): 249,50

## References

- O'Connor, John J.; Robertson, Edmund F.. "MacTutor History of Mathematics archive". University of St Andrews..

## External links

- S at the Mathematics Genealogy Project
- Sergei Natanovich Bernstein and history of approximation theory from Technion — Israel Institute of Technology
- Author profile in the database zbMATH

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