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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 Discrete Chebyshev transform ： ウィキペディア英語版 Discrete Chebyshev transform In applied mathematics, the discrete Chebyshev transform (DCT), named after Pafnuty Chebyshev, is either of two main varieties of DCTs: the discrete Chebyshev transform on the 'roots' grid of the Chebyshev polynomials of the first kind $T\_n\; (x)$ and the discrete Chebyshev transform on the 'extrema' grid of the Chebyshev polynomials of the first kind. ==Discrete Chebyshev transform on the roots grid== The discrete chebyshev transform of u(x) at the points $$ is given by: : $a\_m\; =\backslash frac\backslash sum\_^\; u(x\_n)\; T\_m\; (x\_n)$ where: : $x\_n\; =\; -\backslash cos\backslash left(\backslash frac\; (n+\backslash frac)\backslash right)$ : $a\_m\; =\; \backslash frac\; \backslash sum\_^\; u(x\_n)\; \backslash cos\backslash left(m\; \backslash cos^(x\_n)\backslash right)$ where $p\_m\; =1\; \backslash Leftrightarrow\; m=0$ and $p\_m\; =\; 2$ otherwise. Using the definition of $x\_n$, : $a\_m\; =\backslash frac\; \backslash sum\_^\; u(x\_n)\; \backslash cos\backslash left(\backslash frac(N+n+\backslash frac)\; \backslash right)$ : $a\_m\; =\backslash frac\; \backslash sum\_^\; u(x\_n)\; (-1)^m\backslash cos\backslash left(\backslash frac(n+\backslash frac)\; \backslash right)$ and its inverse transform: : $u\_n\; =\backslash sum\_^\; a\_m\; T\_m\; (x\_n)$ (This so happens to the standard Chebyshev series evaluated on the roots grid.) : $u\_n\; =\backslash sum\_^\; a\_m\; \backslash cos\backslash left(\backslash frac(N+n+\backslash frac)\; \backslash right)$ : $\backslash therefore\; u\_n\; =\backslash sum\_^\; a\_m\; (-1)^m\backslash cos\backslash left(\backslash frac(n+\backslash frac)\; \backslash right)$ This can readily be obtained by manipulating the input arguments to a discrete cosine transform. This can be demonstrated using the following MATLAB code: function a=fct(f,l) %x=-cos(pi/N *((0:N-1)'+1/2)); f=f(end:-1:1,:); A=size(f); N=A(1); if exist('A(3)','var') && A(3)~=1 for i=1:A(3) a(:,:,i)=sqrt(2/N) *dct(f(:,:,i)); a(1,:,i)=a(1,:,i)/sqrt(2); end else a=sqrt(2/N) *dct(f(:,:,i)); a(1,:)=a(1,:)/sqrt(2); end The discrete cosine transform (dct) is in fact computed using a fast Fourier transform algorithm in MATLAB. And the inverse transform is given by the MATLAB code: function f=ifct(a,l) %x=-cos(pi/N *((0:N-1)'+1/2)) k=size(a); N=k(1); a=idct(sqrt(N/2) *(a(2:end,:) )); end 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「Discrete Chebyshev transform」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.