BERNSTEIN CONDITION Meaning and
Definition
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Bernstein condition refers to a mathematical condition or criterion that plays a critical role in several mathematical disciplines, particularly in analysis. It is named after the Russian mathematician Sergei Natanovich Bernstein, who first introduced the concept.
In analysis, the Bernstein condition is a necessary condition that dictates the smoothness of a function or a function's derivative. Specifically, it states that a function is assumed to be infinitely differentiable in a given interval if and only if the sequence of its derivatives decreases faster than an exponential function. This means that the sum of the absolute values of the derivatives multiplied by a positive constant should converge.
The Bernstein condition is often used in approximation theory, where it provides a significant measure of how closely a function can be approximated by a polynomial. It is particularly relevant in the study of Bernstein polynomials, which are used to approximate functions in a wide range of applications such as computer graphics, numerical integration, and data fitting.
In summary, the Bernstein condition is a mathematical criterion that determines the smoothness of a function or a function's derivative. It is an essential concept in analysis and approximation theory, and its applications extend to various mathematical disciplines.
Etymology of BERNSTEIN CONDITION
The word "Bernstein condition" is named after the mathematician Sergei Natanovich Bernstein (1880-1968), who was a prominent Russian mathematician and one of the pioneers in approximation theory. Bernstein made significant contributions to the field of approximation theory, including the formulation of the Bernstein condition.
The Bernstein condition, also known as the Bernstein inequality or the Bernstein-Dzhrbashyan-Matyashevich condition, is a condition that imposes bounds on the growth of the derivatives of a function in terms of its modulus of continuity. It is a fundamental condition for many results in approximation theory, especially in the study of polynomial and rational approximation.
Since Bernstein was instrumental in formulating and studying this condition, it was named after him.
