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 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 ƒ: [0, 2π] → C is an 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 Pn−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 2\pi,

where C(r) depends only on r.

The AkhiezerKreinFavard theorem gives the sharp value of C(r) (called the Akhiezer–Krein–Favard constant):

 C(r) = \frac{4}{\pi} \sum_{k=0}^\infty \frac{(-1)^{k(r+1)}}{(2k+1)^{r+1}}~.

Jackson also proved the following generalisation of Theorem 1:

Theorem 2: Denote by ω(δƒ(r)) the modulus of continuity of the rth derivative of ƒ. Then one can find Pn−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 2\pi

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]