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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Jackson's inequality ： 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 $f:()\backslash to\; \backslash mathbb$ is an $r$ times differentiable periodic function such that $|f^(x)|\; \backslash leq\; 1,\; \backslash quad\; 0\; \backslash leq\; x\; \backslash leq\; 2\backslash pi,$ then, for every positive integer $n$, there exists a trigonometric polynomial $T\_$ of degree at most $n-1$ such that $|f(x)\; -\; T\_(x)|\; \backslash leq\; \backslash frac,\; \backslash quad\; 0\; \backslash leq\; x\; \backslash leq\; 2\backslash pi,$ 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): : $C(r)\; =\; \backslash frac\; \backslash sum\_^\backslash infty\; \backslash frac\}~.$ Jackson also proved the following generalisation of Theorem 1: Theorem 2: Denote by $\backslash omega(\backslash delta,f^)$ the modulus of continuity of the $r$-th derivative of $f$ with the step $\backslash delta$. Then one can find a trigonometric polynomial $T\_n$ of degree $\backslash le\; n$ such that $|f(x)\; -\; T\_n(x)|\; \backslash leq\; \backslash frac,\; \backslash quad\; 0\; \backslash leq\; x\; \backslash leq\; 2\backslash pi.$ 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 $2\backslash pi$-periodic continuous function, there exists a trigonometric polynomial $T\_n$ of degree $\backslash le\; n$ such that $|f(x)-T\_n(x)|\backslash leq\; c(k)\backslash omega\_k\backslash left(\backslash frac,f\backslash right),\backslash quad\; x\backslash in(),$ where constant $c(k)$ depends on $k\backslash in\backslash mathbb$, and $\backslash omega\_k$ 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 $k=2,\; \backslash omega\_2(t,f)\backslash le\; ct,\; t>0$ 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 スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.