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Favard–Akhiezer–Krein theorem : ウィキペディア英語版
Jackson's inequality
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 or modulus of smoothness of the function or 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 f:()\to \mathbb is an r times differentiable periodic function such that
|f^(x)| \leq 1, \quad 0 \leq x \leq 2\pi,
then, for every positive integer n, there exists a trigonometric polynomial T_ of degree at most n-1 such that
|f(x) - T_(x)| \leq \frac, \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 \sum_^\infty \frac}~.
Jackson also proved the following generalisation of Theorem 1:
Theorem 2: Denote by \omega(\delta,f^) the modulus of continuity of the r-th derivative of f with the step \delta. Then one can find a trigonometric polynomial T_n of degree \le n such that
|f(x) - T_n(x)| \leq \frac, \quad 0 \leq x \leq 2\pi.
An even more general result of four authors can be formulated as the following Jackson theorem.
Theorem 3: For every natural number n, if f is 2\pi-periodic continuous function, there exists a trigonometric polynomial T_n of degree \le n such that
|f(x)-T_n(x)|\leq c(k)\omega_k\left(\frac,f\right),\quad x\in(),
where constant c(k) depends on k\in\mathbb, and \omega_k is the k-th order modulus of smoothness.
For k=1 this result was proved by Dunham Jackson. Antoni Zygmund proved the inequality in the case when k=2, \omega_2(t,f)\le ct, t>0 in 1945. Naum Akhiezer proved the theorem in the case k=2 in 1956. For k>2 this result was established by Sergey Stechkin in 1967.

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
ウィキペディアで「Jackson's inequality」の詳細全文を読む



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