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Constructive 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. Example Let ''f'' be a 2''π''-periodic function. Then ''f'' is ''α''- Hölder for some 0 < ''α'' < 1 if and only if for every natural ''n'' there exists a ''Pn'' of degree ''n'' such that : where ''C''(''f'') is a positive number depending on ''f''. The "only if" is due to , see |
Mathematical Analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (mathematics), series, and analytic functions. These theories are usually studied in the context of Real number, real and Complex number, complex numbers and Function (mathematics), functions. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis. Analysis may be distinguished from geometry; however, it can be applied to any Space (mathematics), space of mathematical objects that has a definition of nearness (a topological space) or specific distances between objects (a metric space). History Ancient Mathematical analysis formally developed in the 17th century during the Scientific Revolution, but many of its ideas can be traced back to earlier mathematicians. Early results in analysis were i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Function (mathematics)
In mathematics, a function from a set to a set assigns to each element of exactly one element of .; the words map, mapping, transformation, correspondence, and operator are often used synonymously. The set is called the domain of the function and the set is called the codomain of the function.Codomain ''Encyclopedia of Mathematics'Codomain. ''Encyclopedia of Mathematics''/ref> The earliest known approach to the notion of function can be traced back to works of Persian mathematicians Al-Biruni and Sharaf al-Din al-Tusi. Functions were 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 (that is, they had a high degree of regularity). The concept of a function was formalized at the end of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Approximation Theory
In mathematics, approximation theory is concerned with how function (mathematics), functions can best be approximation, approximated with simpler functions, and with quantitative property, quantitatively characterization (mathematics), characterizing the approximation error, errors introduced thereby. Note that what is meant by ''best'' and ''simpler'' will depend on the application. A closely related topic is the approximation of functions by generalized Fourier series, that is, approximations based upon summation of a series of terms based upon orthogonal polynomials. One problem of particular interest is that of approximating a function in a computer mathematical library, using operations that can be performed on the computer or calculator (e.g. addition and multiplication), such that the result is as close to the actual function as possible. This is typically done with polynomial or Rational function, rational (ratio of polynomials) approximations. The objective is to make t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sergei Bernstein
Sergei Natanovich Bernstein (russian: Серге́й Ната́нович Бернште́йн, sometimes Romanized as ; 5 March 1880 – 26 October 1968) was a Ukrainian and Russian mathematician of Jewish origin known for contributions to partial differential equations, differential geometry, probability theory, and approximation theory. Work Partial differential equations In his doctoral dissertation, submitted in 1904 to Sorbonne, 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 estimates. 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 1920s, he introduced a method for proving limit theorems ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hölder Condition
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 : , f(x) - f(y) , \leq C\, x - y\, ^ for all ''x'' and ''y'' in the domain of ''f''. More generally, the condition can be formulated for functions between any two metric spaces. The number α is called the ''exponent'' of the Hölder condition. A function on an interval satisfying the condition with α > 1 is constant. If α = 1, then the function satisfies a Lipschitz condition. For any α > 0, the condition implies the function is uniformly continuous. The condition is named after Otto Hölder. We have the following chain of strict inclusions for functions over a closed and bounded non-trivial interval of the real line: : Continuously differentiable ⊂ Lipschitz continuous ⊂ α-Hölder continuous ⊂ uniformly continuous ⊂ continuous, where ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trigonometric Polynomial
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. Trigonometric polynomials are widely used, for example in trigonometric interpolation applied to the interpolation of periodic functions. They are used also in the discrete Fourier transform. The term ''trigonometric polynomial'' for the real-valued case can be seen as using the analogy: the functions sin(''nx'') and cos(''nx'') are similar to the monomial basis for polynomials. In the complex case the trigonometric polynomials are spanned by the positive and negative powers of ''e''''ix'', Laurent polynomials in ''z'' under the change of variables ''z'' = ''e''''ix' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dunham Jackson
Dunham Jackson (July 24, 1888 in Bridgewater, Massachusetts – November 6, 1946) 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'' (dated 1941) was reprinted in 2004. Career After attending the local school in Bridgewater, Jackson went up to Harvard in 1904 at the age of 16 to study mathematics, graduating A.B in 1908 and A.M. in 1909. He then moved to continue his studies at Göttingen for two years with the help of Harvard Fellowships. He returned to Harvard in 1911 as an instructor in mathematics and was promoted Assistant Professor in 1916. During the First World War he became an officer in the Ordnance Department where he produced a booklet of range tables for the artillery. In 1919 he took up a professorship in mathematics at the University of Minnesota, remaining t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jackson's Inequality
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. Statement: trigonometric polynomials For trigonometric polynomials, the following was proved by Dunham Jackson: :Theorem 1: If f: ,2\pito \C is an r times differentiable periodic function such that :: \left , f^(x) \right , \leq 1, \qquad x\in ,2\pi :then, for every positive integer n, there exists a trigonometric polynomial T_ of degree at most n-1 such that ::\left , f(x) - T_(x) \right , \leq \frac, \qquad x\in ,2\pi :where C(r) depends only on r. The Akhiezer– Krein– Favard theorem gives the sharp value of C(r) (called the Akhiezer–Krein–Favard constant): : C(r) = \frac \sum_^\infty \frac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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: Let ''f'': , 2π → C be a 2''π''-periodic function, and assume ''r'' is a natural number, and 0 < ''α'' < 1. If there exists a number ''C''(''f'') > 0 and a sequence of s ''n'' ≥ ''n''0 such that : then ''f'' = ''P''''n''0 + ''φ'', where ''φ'' has a bounded ''r''-th derivative which is α-Hölder continuous. See also *[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Approximation Theory
In mathematics, approximation theory is concerned with how function (mathematics), functions can best be approximation, approximated with simpler functions, and with quantitative property, quantitatively characterization (mathematics), characterizing the approximation error, errors introduced thereby. Note that what is meant by ''best'' and ''simpler'' will depend on the application. A closely related topic is the approximation of functions by generalized Fourier series, that is, approximations based upon summation of a series of terms based upon orthogonal polynomials. One problem of particular interest is that of approximating a function in a computer mathematical library, using operations that can be performed on the computer or calculator (e.g. addition and multiplication), such that the result is as close to the actual function as possible. This is typically done with polynomial or Rational function, rational (ratio of polynomials) approximations. The objective is to make t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
