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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 Trigonometric polynomial ： ウィキペディア英語版 Trigonometric polynomial In the mathematical subfields of numerical analysis and mathematical analysis, a trigonometric polynomial is a finite linear combination of functions sin(''nx'') and cos(''nx'') with ''n'' taking on the values of one or more natural numbers. The coefficients may be taken as real numbers, for real-valued functions. For complex coefficients, there is no difference between such a function and a finite Fourier series. Trigonometric polynomials are widely used, for example in trigonometric interpolation applied to the interpolation of periodic functions. They are used also in the discrete Fourier transform. The term ''trigonometric polynomial'' for the real-valued case can be seen as using the analogy: the functions sin(''nx'') and cos(''nx'') are similar to the monomial basis for polynomials. In the complex case the trigonometric polynomials are spanned by the positive and negative powers of ''e''''ix''. ==Formal definition== Any function ''T'' of the form :$T(x)\; =\; a\_0\; +\; \backslash sum\_^N\; a\_n\; \backslash cos\; (nx)\; +\; \backslash mathrm\backslash sum\_^N\; b\_n\; \backslash sin(nx)\; \backslash qquad\; (x\; \backslash in\; \backslash mathbf)$ with ''a''''n'', ''b''''n'' in C for 0 ≤ ''n'' ≤ ''N'', is called a complex trigonometric polynomial of degree ''N'' . Using Euler's formula the polynomial can be rewritten as :$T(x)\; =\; \backslash sum\_^N\; c\_n\; \backslash mathrm^\; \backslash qquad\; (x\; \backslash in\; \backslash mathbf).$ Analogously, letting ''a''''n'', ''b''''n'' be in R, 0 ≤ ''n'' ≤ ''N'' and ''a''''N'' ≠ 0 or ''b''''N'' ≠ 0, then :$t(x)\; =\; a\_0\; +\; \backslash sum\_^N\; a\_n\; \backslash cos\; (nx)\; +\; \backslash sum\_^N\; b\_n\; \backslash sin(nx)\; \backslash qquad\; (x\; \backslash in\; \backslash mathbf)$ is called a real trigonometric polynomial of degree ''N'' . 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「Trigonometric polynomial」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.