Density of the set of generators of wavelet systems

Given a function ψ in L2(Rd), the affine (wavelet) system generated by ψ, associated to an invertible matrix a and a lattice Γ, is the collection of functions {|det a|j/2ψ (ajx - γ) : j ∈ Z, γ ∈ Γ}. In this paper we prove that the set of functions generating affine systems that are a Riesz basis of...

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Autores principales:	Cabrelli, C., Molter, U.M.
Formato:	JOUR
Materias:	Affine systems Riesz basis wavelets Wavelet frames Wavelet set
Acceso en línea:	http://hdl.handle.net/20.500.12110/paper_01764276_v26_n1_p65_Cabrelli
Aporte de:	Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires

id	todo:paper_01764276_v26_n1_p65_Cabrelli
record_format	dspace
spelling	todo:paper_01764276_v26_n1_p65_Cabrelli2023-10-03T15:08:04Z Density of the set of generators of wavelet systems Cabrelli, C. Molter, U.M. Affine systems Riesz basis wavelets Wavelet frames Wavelet set Given a function ψ in L2(Rd), the affine (wavelet) system generated by ψ, associated to an invertible matrix a and a lattice Γ, is the collection of functions {\|det a\|j/2ψ (ajx - γ) : j ∈ Z, γ ∈ Γ}. In this paper we prove that the set of functions generating affine systems that are a Riesz basis of L2(Rd) is dense in L2(Rd). We also prove that a stronger result is true for affine systems that are a frame of L2(Rd). In this case we show that the generators associated to a fixed but arbitrary dilation are a dense set. Furthermore, we analyze the orthogonal case in which we prove that the set of generators of orthogonal (not necessarily complete) affine systems, that are compactly supported in frequency, are dense in the unit sphere of L2(R d) with the induced metric. As a byproduct we introduce the p-Grammian of a function and prove a convergence result of this Grammian as a function of the lattice. This result gives insight in the problem of oversampling of affine systems. © 2007 Springer. Fil:Cabrelli, C. 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_01764276_v26_n1_p65_Cabrelli
institution	Universidad de Buenos Aires
institution_str	I-28
repository_str	R-134
collection	Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA)
topic	Affine systems Riesz basis wavelets Wavelet frames Wavelet set
spellingShingle	Affine systems Riesz basis wavelets Wavelet frames Wavelet set Cabrelli, C. Molter, U.M. Density of the set of generators of wavelet systems
topic_facet	Affine systems Riesz basis wavelets Wavelet frames Wavelet set
description	Given a function ψ in L2(Rd), the affine (wavelet) system generated by ψ, associated to an invertible matrix a and a lattice Γ, is the collection of functions {\|det a\|j/2ψ (ajx - γ) : j ∈ Z, γ ∈ Γ}. In this paper we prove that the set of functions generating affine systems that are a Riesz basis of L2(Rd) is dense in L2(Rd). We also prove that a stronger result is true for affine systems that are a frame of L2(Rd). In this case we show that the generators associated to a fixed but arbitrary dilation are a dense set. Furthermore, we analyze the orthogonal case in which we prove that the set of generators of orthogonal (not necessarily complete) affine systems, that are compactly supported in frequency, are dense in the unit sphere of L2(R d) with the induced metric. As a byproduct we introduce the p-Grammian of a function and prove a convergence result of this Grammian as a function of the lattice. This result gives insight in the problem of oversampling of affine systems. © 2007 Springer.
format	JOUR
author	Cabrelli, C. Molter, U.M.
author_facet	Cabrelli, C. Molter, U.M.
author_sort	Cabrelli, C.
title	Density of the set of generators of wavelet systems
title_short	Density of the set of generators of wavelet systems
title_full	Density of the set of generators of wavelet systems
title_fullStr	Density of the set of generators of wavelet systems
title_full_unstemmed	Density of the set of generators of wavelet systems
title_sort	density of the set of generators of wavelet systems
url	http://hdl.handle.net/20.500.12110/paper_01764276_v26_n1_p65_Cabrelli
work_keys_str_mv	AT cabrellic densityofthesetofgeneratorsofwaveletsystems AT molterum densityofthesetofgeneratorsofwaveletsystems
_version_	1807316135646855168

Density of the set of generators of wavelet systems

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