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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 Walsh function ： ウィキペディア英語版 Walsh function The system of Walsh functions (or, simply, Walsh system) may be viewed as a discrete, digital counterpart of continuous, analog system of trigonometric functions on the unit interval. Unlike trigonometric functions, Walsh functions are only piecewise-continuous and, in fact, are piecewise constant. The functions take the values −1 and +1 only, on sub-intervals defined by dyadic fractions. Both systems form a complete, orthonomal set of functions, an orthonormal basis in Hilbert space $L^2()$ of the square-integrable functions on the unit interval. Both are systems of bounded functions, unlike, say, Haar system or Franklin system. Both trigonometric and Walsh systems admit natural extension by periodicity from the unit interval to the real line $\backslash mathbb\; R$. Furthermore, both Fourier analysis on the unit interval (Fourier series) and on the real line (Fourier transform) have their digital counterparts defined via Walsh system, the Walsh series analogous to the Fourier series, and the Hadamard transform analogous to the Fourier transform. Walsh functions, series, and transforms find various applications in physics and engineering, especially in digital signal processing. They are used in speech recognition, in medical and biological image processing, in digital holography, and other areas. Historically, various numerations of Walsh functions have been used, none of which could be considered particularly superior to another. In what follows, we will use so called Walsh–Paley numeration. ==Definition== We define the sequence of Walsh functions $W\_k:\; ()\; \backslash rightarrow\; \backslash$, $k\; \backslash in\; \backslash mathbb\; N\_0$ as follows. For any $k\; \backslash in\; \backslash mathbb\; N\_0$, $x\; \backslash in\; ()$ let :$k\; =\; \backslash sum\_^\backslash infty\; k\_j2^j,\; k\_j\; \backslash in\; \backslash$,   $x\; =\; \backslash sum\_^\backslash infty\; x\_j2^,\; x\_j\; \backslash in\; \backslash$ such that there are only finitely many non-zero ''k''j and no trailing ''x''j all equal to 1, be the canonical binary representations of integer ''k'' and real number ''x'', correspondingly. Then, by definition :$W\_k(x)\; =\; (-1)^\}$ In particular, $W\_0(x)=1$ everywhere on the interval. Notice that $W\_$ is precisely the Rademacher function ''r''m. Thus, the Rademacher system is a subsystem of the Walsh system. Moreover, every Walsh function is a product of Rademacher functions: :$W\_k(x)\; =\; \backslash prod\_^\backslash infty\; r\_j(x)^$ 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「Walsh function」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.