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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 Bochner–Riesz mean ： ウィキペディア英語版 Bochner–Riesz mean The Bochner–Riesz mean is a summability method often used in harmonic analysis when considering convergence of Fourier series and Fourier integrals. It was introduced by Salomon Bochner as a modification of the Riesz mean. Define : Let $f$ be a periodic function, thought of as being on the n-torus, $\backslash mathbb^n$, and having Fourier coefficients $\backslash hat(k)$ for $k\; \backslash in\; \backslash mathbb^n$. Then the Bochner–Riesz means of complex order $\backslash delta$, $B\_R^\backslash delta\; f$ of (where $R\; >\; 0$ and $\backslash mbox(\backslash delta)\; >\; 0$) are defined as :$B\_R^\backslash delta\; f(\backslash theta)\; =\; \backslash underset\}\; \backslash left(\; 1-\; \backslash frac\; \backslash right)\_+^\backslash delta\; \backslash hat(k)\; e^.$ Analogously, for a function $f$ on $\backslash mathbb^n$ with Fourier transform $\backslash hat(\backslash xi)$, the Bochner–Riesz means of complex order $\backslash delta$, $S\_R^\backslash delta\; f$ (where $R\; >\; 0$ and $\backslash mbox(\backslash delta)\; >\; 0$) are defined as :$S\_R^\backslash delta\; f(x)\; =\; \backslash int\_\; \backslash left(1\; -\; \backslash frac\; \backslash right)\_+^\backslash delta\; \backslash hat(\backslash xi)\; e^\backslash ,d\backslash xi.$ For $\backslash delta\; >\; 0$ and $n=1$, $S\_R^\backslash delta$ and $B\_R^\backslash delta$ may be written as convolution operators, where the convolution kernel is an approximate identity. As such, in these cases, considering the almost everywhere convergence of Bochner–Riesz means for functions in $L^p$ spaces is much simpler than the problem of "regular" almost everywhere convergence of Fourier series/integrals (corresponding to $\backslash delta\; =\; 0$). In higher dimensions, the convolution kernels become more "badly behaved" (specifically, for $\backslash delta\; \backslash leq\; \backslash tfrac$, the kernel is no longer integrable) and establishing almost everywhere convergence becomes correspondingly more difficult. Another question is that of for which $\backslash delta$ and which $p$ the Bochner–Riesz means of an $L^p$ function converge in norm. This is of fundamental importance for $n\; \backslash geq\; 2$, since regular spherical norm convergence (again corresponding to $\backslash delta\; =\; 0$) fails in $L^p$ when $p\; \backslash neq\; 2$. This was shown in a paper of 1971 by Charles Fefferman. By a transference result, the $\backslash mathbb^n$ and $\backslash mathbb^n$ problems are equivalent to one another, and as such, by an argument using the uniform boundedness principle, for any particular $p\; \backslash in\; (1,\; \backslash infty)$, $L^p$ norm convergence follows in both cases for exactly those $\backslash delta$ where $(1-|\backslash xi|^2)^\_+$ is the symbol of an $L^p$ bounded Fourier multiplier operator. For $n=2$, this question has been completely resolved, but for $n\; \backslash geq\; 3$, it has only been partially answered. The case of $n=1$ is not interesting here as convergence follows for $p\; \backslash in\; (1,\; \backslash infty)$ in the most difficult $\backslash delta\; =\; 0$ case as a consequence of the $L^p$ boundedness of the Hilbert transform and an argument of Marcel Riesz. ==References== 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「Bochner–Riesz mean」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.