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, Carlos Alberto, Molter, Ursula Maria
Publicado: 1996
Acceso en línea:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_03342700_v37_n4_p474_Cabrelli
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
id paper:paper_03342700_v37_n4_p474_Cabrelli
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spelling paper:paper_03342700_v37_n4_p474_Cabrelli2023-06-08T15:33:45Z Wavelet transform of the dilation equation Cabrelli, Carlos Alberto Molter, Ursula Maria 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. 1996 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_03342700_v37_n4_p474_Cabrelli 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.
author Cabrelli, Carlos Alberto
Molter, Ursula Maria
spellingShingle Cabrelli, Carlos Alberto
Molter, Ursula Maria
Wavelet transform of the dilation equation
author_facet Cabrelli, Carlos Alberto
Molter, Ursula Maria
author_sort Cabrelli, Carlos Alberto
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
publishDate 1996
url https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_03342700_v37_n4_p474_Cabrelli
http://hdl.handle.net/20.500.12110/paper_03342700_v37_n4_p474_Cabrelli
work_keys_str_mv AT cabrellicarlosalberto wavelettransformofthedilationequation
AT molterursulamaria wavelettransformofthedilationequation
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