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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 Haar wavelet ： ウィキペディア英語版 Haar wavelet In mathematics, the Haar wavelet is a sequence of rescaled "square-shaped" functions which together form a wavelet family or basis. Wavelet analysis is similar to Fourier analysis in that it allows a target function over an interval to be represented in terms of an orthonormal function basis. The Haar sequence is now recognised as the first known wavelet basis and extensively used as a teaching example. The Haar sequence was proposed in 1909 by Alfréd Haar.〔see p. 361 in .〕 Haar used these functions to give an example of an orthonormal system for the space of square-integrable functions on the unit interval (). The study of wavelets, and even the term "wavelet", did not come until much later. As a special case of the Daubechies wavelet, the Haar wavelet is also known as Db1. The Haar wavelet is also the simplest possible wavelet. The technical disadvantage of the Haar wavelet is that it is not continuous, and therefore not differentiable. This property can, however, be an advantage for the analysis of signals with sudden transitions, such as monitoring of tool failure in machines. The Haar wavelet's mother wavelet function $\backslash psi(t)$ can be described as : $\backslash psi(t)\; =\; \backslash begin$ 1 \quad & 0 \leq t < \frac,\\ -1 & \frac \leq t < 1,\\ 0 &\mbox \end Its scaling function $\backslash phi(t)$ can be described as : == Haar functions and Haar system == For every pair ''n'', ''k'' of integers in Z, the Haar function ψ''n'', ''k'' is defined on the real line R by the formula :$\backslash psi\_(t)\; =\; 2^\; \backslash psi(2^n\; t-k),\; \backslash quad\; t\; \backslash in\; \backslash mathbf.$ This function is supported on the right-open interval , ''i.e.'', it vanishes outside that interval. It has integral 0 and norm 1 in the Hilbert space ''L''2(R), :$\backslash int\_(t)\; \backslash ,\; d\; t\; =\; 0,\; \backslash quad\; \backslash |\backslash psi\_\backslash |^2\_\; =\; \backslash int\_(t)^2\; \backslash ,\; d\; t\; =\; 1.$ The Haar functions are pairwise orthogonal, :$\backslash int\_(t)\; \backslash psi\_(t)\; \backslash ,\; d\; t\; =\; \backslash delta\_\; \backslash delta\_,$ where ''δ''i,j represents the Kronecker delta. Here is the reason for orthogonality: when the two supporting intervals $I\_$ and $I\_$ are not equal, then they are either disjoint, or else, the smaller of the two supports, say $I\_$, is contained in the lower or in the upper half of the other interval, on which the function $\backslash psi\_$ remains constant. It follows in this case that the product of these two Haar functions is a multiple of the first Haar function, hence the product has integral 0. The Haar system on the real line is the set of functions :$\backslash ,\; \backslash ;\; k\; \backslash in\; \backslash mathbf\; \backslash \}.$ It is complete in ''L''2(R): ''The Haar system on the line is an orthonormal basis in'' ''L''2(R). 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「Haar wavelet」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.