Fujiki class C

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, 2Potter or Ceramic Artist Harry Rave from Cobden, spends time with hobbies for instance magic, property developers house in singapore singapore and fitness. Finds inspiration through travel and just spent 7 months at Keoladeo National Park.] → C is an r times differentiable periodic function such that

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

then, for every natural n, there exists a trigonometric polynomial P_n−1 of degree at most n − 1 such that

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

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):

$C(r)=\frac{4}{\pi }\sum _{k=0}^{\mathrm{\infty }}\frac{(-1{)}^{k(r+1)}}{(2k+1{)}^{r+1}}.$

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

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

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

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