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 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 ƒ: [0, 2π] → C is an r times differentiable periodic function such that
then, for every natural n, there exists a trigonometric polynomial Pn−1 of degree at most n − 1 such that
where C(r) depends only on r.
Jackson also proved the following generalisation of Theorem 1:
Theorem 2: Denote by ω(δ, ƒ(r)) the modulus of continuity of the rth derivative of ƒ. Then one can find Pn−1 such that
- 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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