In
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. Note that what is meant by ''best'' and ''simpler'' wil ...
, Bernstein's theorem is a converse to
Jackson's theorem. The first results of this type were proved by
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 parti ...
in 1912.
For approximation by
trigonometric polynomials, the result is as follows:
Let ''f'':
, 2π
The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
→ C be a 2''π''-
periodic function
A periodic function is a function that repeats its values at regular intervals. For example, the trigonometric functions, which repeat at intervals of 2\pi radians, are periodic functions. Periodic functions are used throughout science to d ...
, and assume ''r'' is a
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called '' cardinal ...
, and 0 < ''α'' < 1. If there exists a number ''C''(''f'') > 0 and a sequence of
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 c ...
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
*
Bernstein's lethargy theorem
*
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 Bernste ...
References
Theorems in approximation theory
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