Adaptive Wavelet-Analysis: Theorie und Software by Mladen Victor Wickerhauser, Kurt Jetter

By Mladen Victor Wickerhauser, Kurt Jetter

Professor Dr. Mladen Victor Wickerhauser lehrt an der collage in St. Louis, Missouri, USA.

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Extra resources for Adaptive Wavelet-Analysis: Theorie und Software

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Es sei T eine temperierte Distribution, und fUr eine Funktion ¢ der Schwartz-Klasse sei ¢y{x) ¢(x + y). Man zeige fUr tf;{y) T{¢y), daB tf; auch eine Funktion der Schwartz-Klasse ist. = = 3. , man zeige (V2cos21Tnx,v'isin21Tmx) = 0, (V2 cos 21Tnx, V2 cos 21Tmx) (V2 sin 21Tnx, v'i sin 21Tmx) 11111 = 1Iv'i sin 21Tnxll 0, fUr aIle n, mE Z, n 0, fUr fUr alle n, mE Z; :I m; aIle n, m E Z, n :I m; 1Iv'i cos 21Tnxll = 1. und 4. Man zeige: 1st f = gin L2([0,1]) und sind fund 9 stetig, so gilt f(x) = g(x) fUr jedes x.

31 1st lIull endlich, so sind auch die Werte IldptyUIi fur y endlich, und es gilt = 0,1, ... 4 Approximation In jeder numerischen Rechnung konnen wir nur endlich viele Parameter verwenden. Eine Funktion einer reellen Variablen muE deshalb erst approximiert werden durch ihre Projektion auf einen endlich-dimensionalen Raum, der sich als lineare Hiille einer endlichen Menge ~ N = {¢> N,k (t) : k = 0, 1, 2, ... , N - I} von sogenannten Synthese-Funktionen ergibt. } sein. ) = q) bezeichnet hierbei den grofiten gemeinsamen Teiler der natiirlichen Zahlen p und q.

Ein Funktional oder eine MaEzahl, die fUr U berechnet wird und die nicht von dieser Wahl des Ursprungs abhangt, muE fiir jede Folge TyU, unabhangig von y, den gleichen Wert ergeben. Zum Beispiel hangt die Energie IIuI1 2 einer Folge nicht von der Wahl des Ursprungs ab: Fiir aIle y gilt (1. 72) Diese Invarianz kann benutzt werden, urn Formeln fiir die Berechnung der MaEzahl algebraisch zu vereinfachen. 1m allgemeinen kommutieren Translation und Dilatation nicht miteinander; es gibt aber eine Verkniipfungsrelation: Fiir aIle x, y, p gilt (1.

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