Jackson's inequality

In approximation theory, Jackson's inequality is an inequality (proved by Dunham Jackson) between the value of function's best approximation by polynomials and the modulus of continuity of its derivatives. Here is one of the simple cases (concerning approximation by trigonometric polynomials):

Theorem: If $f: [0, 2\pi] \to \mathbb{C}$ is a r times differentiable periodic function such that

$|f^{(r)}(x)| \leq 1, \quad 0 \leq x \leq 2\pi,$

then, for every natural $n$ , there exists a trigonometric polynomial $P n - 1$ of degree at most $n - 1$ such that

$|f(x) - P_{n-1}(x)| \leq C(r)/n^r, \quad 0 \leq x \leq 2\pi,$

where $C (r)$ depends only on $r$ .

A more general fact:

Theorem Denote by $ω(δ, f (r))$ the modulus of continuity of the $r$ -th derivative of $f$ . Then one can find $P n - 1$ such that

$|f(x) - P_{n-1}(x)| \leq C_1(r) \omega(1/n, f^{(r)}) / n^r, \quad 0 \leq x \leq 2\pi$

Generalisations and extensions are called Jackson-type theorems. See also Bernstein-type theorems for reverse results.

[edit] External links

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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