Jackson's inequality

Summary

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.[1] Informally speaking, the smoother the function is, the better it can be approximated by polynomials.

Statement: trigonometric polynomialsEdit

For trigonometric polynomials, the following was proved by Dunham Jackson:

Theorem 1: If   is an   times differentiable periodic function such that
 
then, for every positive integer  , there exists a trigonometric polynomial   of degree at most   such that
 
where   depends only on  .

The AkhiezerKreinFavard theorem gives the sharp value of   (called the Akhiezer–Krein–Favard constant):

 

Jackson also proved the following generalisation of Theorem 1:

Theorem 2: One can find a trigonometric polynomial   of degree   such that
 
where   denotes the modulus of continuity of function   with the step  

An even more general result of four authors can be formulated as the following Jackson theorem.

Theorem 3: For every natural number  , if   is  -periodic continuous function, there exists a trigonometric polynomial   of degree   such that
 
where constant   depends on   and   is the  -th order modulus of smoothness.

For   this result was proved by Dunham Jackson. Antoni Zygmund proved the inequality in the case when   in 1945. Naum Akhiezer proved the theorem in the case   in 1956. For   this result was established by Sergey Stechkin in 1967.

Further remarksEdit

Generalisations and extensions are called Jackson-type theorems. A converse to Jackson's inequality is given by Bernstein's theorem. See also constructive function theory.

ReferencesEdit

  1. ^ Achieser, N.I. (1956). Theory of Approximation. New York: Frederick Ungar Publishing Co.

External linksEdit