Constructive function theory

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In mathematical analysis, constructive function theory is a field which studies the connection between the smoothness of a function and its degree of approximation. [1] [2] It is closely related to approximation theory. The term was coined by Sergei Bernstein.

Mathematical analysis branch of pure mathematics

Mathematical analysis is the branch of mathematics dealing with limits and related theories, such as differentiation, integration, measure, infinite series, and analytic functions.

Function (mathematics) Mathematical binary relation

In mathematics, a function was originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a function of time. Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable. The concept of function was formalized at the end of the 19th century in terms of set theory, and this greatly enlarged the domains of application of the concept.

Approximation theory Theory of getting acceptably close inexact mathematical calculations

In mathematics, approximation theory is concerned with how functions can best be approximated with simpler functions, and with quantitatively characterizing the errors introduced thereby. Note that what is meant by best and simpler will depend on the application.

Contents

Example

Let f be a 2π-periodic function. Then f is α-Hölder for some 0 < α < 1 if and only if for every natural n there exists a trigonometric polynomial Pn of degree n such that

In mathematics, a real or complex-valued function f on d-dimensional Euclidean space satisfies a Hölder condition, or is Hölder continuous, when there are nonnegative real constants C, α>0, such that

In the mathematical subfields of numerical analysis and mathematical analysis, a trigonometric polynomial is a finite linear combination of functions sin(nx) and cos(nx) with n taking on the values of one or more natural numbers. The coefficients may be taken as real numbers, for real-valued functions. For complex coefficients, there is no difference between such a function and a finite Fourier series.

where C(f) is a positive number depending on f. The "only if" is due to Dunham Jackson, see Jackson's inequality; the "if" part is due to Sergei Bernstein, see Bernstein's theorem (approximation theory).

Dunham Jackson was a mathematician who worked within approximation theory, notably with trigonometrical and orthogonal polynomials. He is known for Jackson's inequality. He was awarded the Chauvenet Prize in 1935. His book Fourier Series and Orthogonal Polynomials was reprinted in 2004.

In approximation theory, Jackson's inequality is an inequality bounding the value of function's best approximation by algebraic or trigonometric polynomials in terms of the modulus of continuity or modulus of smoothness of the function or of its derivatives. Informally speaking, the smoother the function is, the better it can be approximated by polynomials.

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.

Notes

  1. ↑ "Constructive Theory of Functions".
  2. ↑ Telyakovskii, S.A. (2001) [1994], "Constructive theory of functions", in Hazewinkel, Michiel, Encyclopedia of Mathematics , Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN   978-1-55608-010-4

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References

Naum Akhiezer Russian mathematician

Naum Ilyich Akhiezer was a Soviet mathematician of Jewish origin, known for his works in approximation theory and the theory of differential and integral operators. He is also known as the author of classical books on various subjects in analysis, and for his work on the history of mathematics. He is the brother of the theoretical physicist Aleksander Akhiezer.

Isidor Pavlovich Natanson was a Swiss-born Soviet mathematician known for contributions to real analysis and constructive function theory, in particular, for his textbooks on these subjects. His son, Garal'd Natanson (1930–2003), was also a known mathematician.

Mathematical Reviews is a journal published by the American Mathematical Society (AMS) that contains brief synopses, and in some cases evaluations, of many articles in mathematics, statistics, and theoretical computer science. The AMS also publishes an associated online bibliographic database called MathSciNet which contains an electronic version of Mathematical Reviews and additionally contains citation information for over 3.5 million items as of 2018.

Natanson, I. P. (1965). Constructive function theory. Vol. II. Approximation in mean. New York: Frederick Ungar Publishing Co. MR   0196341. 
Natanson, I. P. (1965). Constructive function theory. Vol. III. Interpolation and approximation quadratures. New York: Ungar Publishing Co. MR   0196342. 
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