Journal article

Publication year: 2007

Pages: 28-46

Journal: Journal of Approximation Theory

Volume number: 147

Issue number: 1

ISSN: 0021-9045

Open access status: Bronze

DOI Link: https://doi.org/10.1016/j.jat.2006.12.011

Publisher: Elsevier

Abstract: 
An irregular wavelet frame has the form Psi (Psi, Lambda) ={a(-1/2) Psi(x/a-b)}(a,b)epsilon Lambda where Psi epsilon L-2 (R) and Lambda is an arbitrary sequence of points in the affine group A R+ x R. Such irregular wavelet frames are poorly understood, yet they arise naturally, e.g.. from sampling theory or the inevitability of perturbations. This paper proves that irregular wavelet frames satisfy a Homogeneous Approximation Property, which essentially states that the rate of approximation of a wavelet frame expansion of a function f is invariant under time-scale shifts off, even though A is not required to have any structure-it is only required that the wavelet 0 has a modest amount of time-scale concentration. It is shown that the Homogeneous Approximation Property has several implications oil the geometry of A, and in particular a relationship between the affine Beurling density of the frame and the affine Beurling density of any other Riesz basis of wavelets is derived. This further yields necessary conditions for the existence of wavelet frames, and insight into the fundamental question of why there is no Nyquist density phenomenon for wavelet frames, as there is for Gabor frames that are generated from time-frequency shifts. (C) 2007 Elsevier Inc. All rights reserved.

Harvard Citation style: Heil, C. and Kutyniok, G. (2007) The Homogeneous Approximation Property for wavelet frames, Journal of Approximation Theory, 147(1), pp. 28-46. https://doi.org/10.1016/j.jat.2006.12.011

APA Citation style: Heil, C., & Kutyniok, G. (2007). The Homogeneous Approximation Property for wavelet frames. Journal of Approximation Theory. 147(1), 28-46. https://doi.org/10.1016/j.jat.2006.12.011