Non-Lebesgue multiresolution analyses

Lawrence Baggett¹

¹ Department of Mathematics University of Colorado UCB 395, Boulder, CO 80309, U.S.A.

Colloquium Mathematicum, Tome 118 (2010) no. 1, pp. 133-145

Résumé

Classical notions of wavelets and multiresolution analyses deal with the Hilbert space $L^2(\mathbb R)$ and the standard translation and dilation operators. Key in the study of these subjects is the low-pass filter, which is a periodic function $h \in L^2([0,1))$ that satisfies the classical quadrature mirror filter equation $|h(x)|^2+|h(x+1/2)|^2 =2.$ This equation is satisfied almost everywhere with respect to Lebesgue measure on the torus. Generalized multiresolution analyses and wavelets exist in abstract Hilbert spaces with more general translation and dilation operators. Moreover, the concept of the low-pass filter has been generalized in various ways. It may be a matrix-valued function, it may not satisfy any obvious analog of a filter equation, and it may be an element of a non-Lebesgue $L^2$ space. In this article we discuss the last of these generalizations, i.e., filters that are elements of non-Lebesgue $L^2$ spaces. We give examples of such filters, and we derive ageneralization of the filter equation.

Détail

DOI : 10.4064/cm118-1-6

Mots-clés : classical notions wavelets multiresolution analyses hilbert space mathbb standard translation dilation operators key study these subjects low pass filter which periodic function satisfies classical quadrature mirror filter equation equation satisfied almost everywhere respect lebesgue measure torus generalized multiresolution analyses wavelets exist abstract hilbert spaces general translation dilation operators moreover concept low pass filter has generalized various ways may matrix valued function may satisfy obvious analog filter equation may element non lebesgue space article discuss these generalizations filters elements non lebesgue spaces examples filters derive ageneralization filter equation