Sergei Natanovich Bernstein
Encyclopedia
Sergei Natanovich Bernstein ' onMouseout='HidePop("28409")' href="/topics/Odessa">Odessa
– October 26, 1968, Moscow
) was a Russia
n and Soviet mathematician
known for contributions to partial differential equations, differential geometry, probability theory
, and approximation theory
.
, 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 estimate
s.
In the 1920-s, he introduced a method for proving limit theorems
for sums of dependent random variable
s.
, a field studying the connection between smoothness properties of a function and its approximations by polynomials. In particular, he proved Bernstein's theorem (approximation theory)
.
Odessa
Odessa or Odesa is the administrative center of the Odessa Oblast located in southern Ukraine. The city is a major seaport located on the northwest shore of the Black Sea and the fourth largest city in Ukraine with a population of 1,029,000 .The predecessor of Odessa, a small Tatar settlement,...
– October 26, 1968, Moscow
Moscow
Moscow is the capital, the most populous city, and the most populous federal subject of Russia. The city is a major political, economic, cultural, scientific, religious, financial, educational, and transportation centre of Russia and the continent...
) was a Russia
Russia
Russia or , officially known as both Russia and the Russian Federation , is a country in northern Eurasia. It is a federal semi-presidential republic, comprising 83 federal subjects...
n and Soviet mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
known for contributions to partial differential equations, differential geometry, probability theory
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
, and approximation theory
Approximation theory
In mathematics, approximation theory is concerned with how functions can best be approximated with simpler functions, and with quantitatively characterizing the errors introduced thereby...
.
Partial differential equations
In his doctoral dissertation, submitted in 1904 to the SorbonneUniversity of Paris
The University of Paris was a university located in Paris, France and one of the earliest to be established in Europe. It was founded in the mid 12th century, and officially recognized as a university probably between 1160 and 1250...
, Bernstein solved Hilbert's nineteenth problem
Hilbert's nineteenth problem
Hilbert's nineteenth problem is one of the 23 Hilbert problems set out in a celebrated list compiled in 1900 by David Hilbert. It asks whether the solutions of regular problems in the calculus of variations are always analytic.-History:...
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 estimate
A priori estimate
In the theory of partial differential equations, an a priori estimate is an estimate for the size of a solution or its derivatives of a partial differential equation. A priori is Latin for "from before" and refers to the fact that the estimate for the solution is derived before the solution is...
s.
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 1920-s, he introduced a method for proving limit theorems
Central limit theorem
In probability theory, the central limit theorem states conditions under which the mean of a sufficiently large number of independent random variables, each with finite mean and variance, will be approximately normally distributed. The central limit theorem has a number of variants. In its common...
for sums of dependent random variable
Random variable
In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...
s.
Approximation theory
Bernstein laid the foundations of constructive function theoryConstructive function theory
In mathematical analysis, constructive function theory is a field which studies the connection between the smoothness of a function and its degree of approximation. It is closely related to approximation theory. The term was coined by Sergei Bernstein....
, a field studying the connection between smoothness properties of a function and its approximations by polynomials. In particular, he proved Bernstein's theorem (approximation theory)
Bernstein's theorem (approximation theory)
In approximation theory, Bernstein's theorem is a converse to Jackson's theorem. The first results of this type were proved by Sergei Bernstein in 1912.For approximation by trigonometric polynomials, the result is as follows:...
.
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
See also
- A priori estimateA priori estimateIn the theory of partial differential equations, an a priori estimate is an estimate for the size of a solution or its derivatives of a partial differential equation. A priori is Latin for "from before" and refers to the fact that the estimate for the solution is derived before the solution is...
- Bernstein algebra
- Bernstein's inequality (mathematical analysis)
- Bernstein inequalities in probability theory
- Bernstein polynomialBernstein polynomialIn the mathematical field of numerical analysis, a Bernstein polynomial, named after Sergei Natanovich Bernstein, is a polynomial in the Bernstein form, that is a linear combination of Bernstein basis polynomials....
- Bernstein's problemBernstein's problemIn differential geometry, Bernstein's problem is as follows: if the graph of a function on Rn−1 is a minimal surface in Rn, does this imply that the function is linear?...
- Bernstein's theorem (approximation theory)Bernstein's theorem (approximation theory)In approximation theory, Bernstein's theorem is a converse to Jackson's theorem. The first results of this type were proved by Sergei Bernstein in 1912.For approximation by trigonometric polynomials, the result is as follows:...
- Bernstein's theorem on monotone functions
- Bernstein–von Mises theoremBernstein–von Mises theoremIn Bayesian inference, the Bernstein–von Mises theorem provides the basis for the important result that the posterior distribution for unknown quantities in any problem is effectively independent of the prior distribution once the amount of information supplied by a sample of data is large...