- Jackson's inequality
In
approximation theory , Jackson's inequality is an inequality (proved byDunham Jackson ) between the value of function's best approximation by polynomials and themodulus of continuity of its derivatives. Here is one of the simple cases (concerning approximation bytrigonometric polynomials ):Theorem: If f: [0, 2pi] o mathbb{C} is a "r" times differentiable
periodic function such that: f^{(r)}(x)| leq 1, quad 0 leq x leq 2pi,
then, for every natural "n", there exists a
trigonometric polynomial "P""n"−1 of degree at most "n" − 1 such that: f(x) - P_{n-1}(x)| leq frac{C(r)}{n^r}, quad 0 leq x leq 2pi,
where "C"("r") depends only on "r".
A more general fact:
Theorem Denote by omega(delta, f^{(r)}) the
modulus of continuity of the "r"th derivative of "ƒ". Then one can find "P""n"−1 such that: f(x) - P_{n-1}(x)| leq frac{C_1(r) omega(1/n, f^{(r)})}{n^r}, quad 0 leq x leq 2pi
Generalisations and extensions are called
Jackson-type theorems . See alsoBernstein-type theorems for reverse results.External links
* [http://eom.springer.de/J/j054000.htm Jackson inequality] on Encyclopaedia of Mathematics.
References
* N.I.Achiezer (Akhiezer), Theory of approximation, Translated by Charles J. Hyman Frederick Ungar Publishing Co., New York 1956 x+307 pp.
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