Wavelet transform of the dilation equation

In this article we study the dilation equation f(x) = ∑hchf(2x-h) in ℒ2(ℝ) using a wavelet approach. We see that the structure of Multiresolution Analysis adapts very well to the study of scaling functions. The equation is reduced to an equation in a subspace of ℒ2(ℝ) of much lower resolution. This...

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Autores principales: Cabrelli, C.A., Molter, U.M.
Formato: JOUR
Acceso en línea:http://hdl.handle.net/20.500.12110/paper_03342700_v37_n4_p474_Cabrelli
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Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires
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spelling todo:paper_03342700_v37_n4_p474_Cabrelli2023-10-03T15:25:19Z Wavelet transform of the dilation equation Cabrelli, C.A. Molter, U.M. In this article we study the dilation equation f(x) = ∑hchf(2x-h) in ℒ2(ℝ) using a wavelet approach. We see that the structure of Multiresolution Analysis adapts very well to the study of scaling functions. The equation is reduced to an equation in a subspace of ℒ2(ℝ) of much lower resolution. This simpler equation is then "wavelet transformed" to obtain a discrete dilation equation. In particular we study the case of compactly supported solutions and we see that conditions for the existence of solutions are given by convergence of infinite products of matrices. These matrices are of the type obtained by Daubechies, and, when the analyzing wavelet is the Haar wavelet, they are exactly the same. © Australian Mathematical Society, 1996. Fil:Cabrelli, C.A. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. Fil:Molter, U.M. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. JOUR info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar http://hdl.handle.net/20.500.12110/paper_03342700_v37_n4_p474_Cabrelli
institution Universidad de Buenos Aires
institution_str I-28
repository_str R-134
collection Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA)
description In this article we study the dilation equation f(x) = ∑hchf(2x-h) in ℒ2(ℝ) using a wavelet approach. We see that the structure of Multiresolution Analysis adapts very well to the study of scaling functions. The equation is reduced to an equation in a subspace of ℒ2(ℝ) of much lower resolution. This simpler equation is then "wavelet transformed" to obtain a discrete dilation equation. In particular we study the case of compactly supported solutions and we see that conditions for the existence of solutions are given by convergence of infinite products of matrices. These matrices are of the type obtained by Daubechies, and, when the analyzing wavelet is the Haar wavelet, they are exactly the same. © Australian Mathematical Society, 1996.
format JOUR
author Cabrelli, C.A.
Molter, U.M.
spellingShingle Cabrelli, C.A.
Molter, U.M.
Wavelet transform of the dilation equation
author_facet Cabrelli, C.A.
Molter, U.M.
author_sort Cabrelli, C.A.
title Wavelet transform of the dilation equation
title_short Wavelet transform of the dilation equation
title_full Wavelet transform of the dilation equation
title_fullStr Wavelet transform of the dilation equation
title_full_unstemmed Wavelet transform of the dilation equation
title_sort wavelet transform of the dilation equation
url http://hdl.handle.net/20.500.12110/paper_03342700_v37_n4_p474_Cabrelli
work_keys_str_mv AT cabrellica wavelettransformofthedilationequation
AT molterum wavelettransformofthedilationequation
_version_ 1807322226385485824

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