# Sergei Natanovich Bernstein

**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]

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

Sergei Natanovich Bernstein | |

Born | |

Died | 26 October 1968 88) | (aged

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 |

Scientific career | |

Fields | Mathematics |

Institutions | University of Paris University of Göttingen University of Kharkiv Leningrad University Steklov Institute of Mathematics |

Doctoral advisor | Charles Émile Picard David Hilbert |

Doctoral students | Yakov Geronimus Sergey Stechkin |

## 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the first axiomatic foundation of probability theory, based on the underlying algebraic structure.[4] It was later superseded by the measure-theoretic approach of Kolmogorov.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*. - Lozinskii, S. M. (1983). "On the hundredth anniversary of the birth of S. N. Bernstein".
*Russ. Math. Surv*.**38**(3): 163. doi:10.1070/RM1983v038n03ABEH003497. - Akhiezer, N.I.; Petrovskii, I.G. (1961). "S. N. Bernshtein's contribution to the theory of partial differential equations".
*Russ. Math. Surv*.**16**(2): 1–15. doi:10.1070/RM1961v016n02ABEH004101. - Linnik, Ju. V. (1961). "The contribution of S. N. Bernšteĭn to the theory of probability".
*Russ. Math. Surv*.**16**(2): 21–22. doi:10.1070/rm1961v016n02abeh004103. MR 0130818. - Videnskii, V. S. (1961). "Sergei Natanovich Bernshtein — founder of the constructive theory of functions".
*Russ. Math. Surv*.**16**: 17. doi: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

## External links

- Sergei Natanovich Bernstein 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