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 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 .
Jackson also proved the following generalisation of Theorem 1:
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.
- Achieser, N.I. (1956). Theory of Approximation. New York: Frederick Ungar Publishing Co.
- Korneichuk, N.P.; Motornyi, V.P. (2001), "Jackson_inequality", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
- Weisstein, Eric W. "Jackson's Theorem". MathWorld.
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