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Abstract:

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.

Registro:

Documento: Artículo
Título:Density of the set of generators of wavelet systems
Autor:Cabrelli, C.; Molter, U.M.
Filiación:Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, 1428 Capital Federal, Argentina
CONICET, Argentina
Palabras clave:Affine systems; Riesz basis wavelets; Wavelet frames; Wavelet set
Año:2007
Volumen:26
Número:1
Página de inicio:65
Página de fin:81
DOI: http://dx.doi.org/10.1007/s00365-006-0644-5
Título revista:Constructive Approximation
Título revista abreviado:Constr. Approx.
ISSN:01764276
Registro:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_01764276_v26_n1_p65_Cabrelli

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Citas:

---------- APA ----------
Cabrelli, C. & Molter, U.M. (2007) . Density of the set of generators of wavelet systems. Constructive Approximation, 26(1), 65-81.
http://dx.doi.org/10.1007/s00365-006-0644-5
---------- CHICAGO ----------
Cabrelli, C., Molter, U.M. "Density of the set of generators of wavelet systems" . Constructive Approximation 26, no. 1 (2007) : 65-81.
http://dx.doi.org/10.1007/s00365-006-0644-5
---------- MLA ----------
Cabrelli, C., Molter, U.M. "Density of the set of generators of wavelet systems" . Constructive Approximation, vol. 26, no. 1, 2007, pp. 65-81.
http://dx.doi.org/10.1007/s00365-006-0644-5
---------- VANCOUVER ----------
Cabrelli, C., Molter, U.M. Density of the set of generators of wavelet systems. Constr. Approx. 2007;26(1):65-81.
http://dx.doi.org/10.1007/s00365-006-0644-5