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 (m ...

, constructive function theory is a field which studies the connection between the smoothness of a

function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-orie ...

and its degree of

approximation An approximation is anything that is intentionally similar but not exactly equal to something else. Etymology and usage The word ''approximation'' is derived from Latin ''approximatus'', from ''proximus'' meaning ''very near'' and the prefix '' ...

. It is closely related to

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 ...

. The term was coined 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 ...

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

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 ...

''P_n'' of degree ''n'' such that :

\max_ ,  f(x) - P_n(x) ,  \leq \frac,

where ''C''(''f'') is a positive number depending on ''f''. The "only if" is due to Dunham Jackson, see

Jackson's inequality In approximation theory, Jackson's inequality is an inequality bounding the value of function's best approximation by polynomials, algebraic or trigonometric polynomials in terms of the modulus of continuity or modulus of smoothness of the function ...

; the "if" part is due to

, see Bernstein's theorem (approximation theory).

Notes

References

* * : : {{cite book, mr=0196342, last=Natanson, first=I. P., author-link=Isidor Natanson, title=Constructive function theory. Vol. III. Interpolation and approximation quadratures, publisher=Ungar Publishing Co., location=New York, year=1965 Approximation theory Smooth functions