Non-Lebesgue multiresolution analyses

Lawrence Baggett. "Non-Lebesgue multiresolution analyses." Colloquium Mathematicae 118.1 (2010): 133-145. <http://eudml.org/doc/284355>.

@article{LawrenceBaggett2010,
abstract = {Classical notions of wavelets and multiresolution analyses deal with the Hilbert space L²(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 ∈ L²([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 L² space. In this article we discuss the last of these generalizations, i.e., filters that are elements of non-Lebesgue L² spaces. We give examples of such filters, and we derive ageneralization of the filter equation.},
author = {Lawrence Baggett},
journal = {Colloquium Mathematicae},
keywords = {multiresolution analysis; filters; filter equations; Ruelle operator},
language = {eng},
number = {1},
pages = {133-145},
title = {Non-Lebesgue multiresolution analyses},
url = {http://eudml.org/doc/284355},
volume = {118},
year = {2010},
}

TY - JOUR
AU - Lawrence Baggett
TI - Non-Lebesgue multiresolution analyses
JO - Colloquium Mathematicae
PY - 2010
VL - 118
IS - 1
SP - 133
EP - 145
AB - Classical notions of wavelets and multiresolution analyses deal with the Hilbert space L²(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 ∈ L²([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 L² space. In this article we discuss the last of these generalizations, i.e., filters that are elements of non-Lebesgue L² spaces. We give examples of such filters, and we derive ageneralization of the filter equation.
LA - eng
KW - multiresolution analysis; filters; filter equations; Ruelle operator
UR - http://eudml.org/doc/284355
ER -