Jackson's inequality

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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 polynomials[edit]

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: Denote by the modulus of continuity of the -th derivative of with the step . Then one can find a trigonometric polynomial of degree such that

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 remarks[edit]

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.

References[edit]

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

External links[edit]