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

For trigonometric polynomials, the following was proved by Dunham Jackson:

Theorem 1: If ƒ: [0, 2 $π$ ] → 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 P_n−1 of degree at most n − 1 such that

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

where C(r) depends only on r.

The Akhiezer–Krein–Favard 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(δ, ƒ^(r)) the modulus of continuity of the rth derivative of ƒ. Then one can find P_n−1 such that

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

Further remarks

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

↑ Achieser, N.I. (1956). Theory of Approximation. New York: Frederick Ungar Publishing Co.

External links

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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