Frames in spaces with finite rate of innovation
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Frames in spaces with finite rate of innovation Qiyu Sun
Received: 14 February 2006 / Accepted: 15 May 2006 / Published online: 18 October 2006 © Springer Science + Business Media B.V. 2006
Abstract Signals with finite rate of innovation are those signals having finite degrees of freedom per unit of time that specify them. In this paper, we introduce a prototypical space Vq (8, 3) modeling signals with finite rate of innovation, such as stream of (different) pulses found in GPS applications, cellular radio and ultra wideband communication. In particular, the space Vq (8, 3) is generated by a family of well-localized molecules 8 of similar size located on a relatively separated set 3 using `q coefficients, and hence is locally finitely generated. Moreover that space Vq (8, 3) includes finitely generated shift-invariant spaces, spaces of non-uniform splines, and the twisted shift-invariant space in Gabor (Wilson) system as its special cases. Use the well-localization property of the generator 8, we show that if the generator 8 is a frame for the space V2 (8, 3) and has polynomial (sub-exponential) decay, then its canonical dual (tight) frame has the same polynomial (sub-exponential) decay. We apply the above result about the canonical dual frame to the study of the Banach frame property of the generator 8 for the space Vq (8, 3) with q 6 = 2, and of the polynomial (sub-exponential) decay property of the mask associated with a refinable function that has polynomial (sub-exponential) decay. Keywords frame · Banach frame · localized frame · signals with finite rate of innovation · space of homogenous type · matrix algebra · refinable function · wavelets Mathematics Subject Classifications (2000) Primary 42C40 · Secondary 47B37 · 46C07
Q. Sun (B) Department of Mathematics, University of Central Florida, Orlando, FL 32816, USA e-mail: [email protected]
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Adv Comput Math (2008) 28:301–329
1. Introduction A signal that has finite degree of freedom per unit of time, the number of samples per unit of time that specify it, is called to be a signal with finite rate of innovation [60]. In this paper, we introduce a prototypical space generated by a family of separated–located molecules of similar size for modeling (periodic, discrete) signals with finite rate of innovation, and study the (Banach) frame property of that generating family for that prototypical space. The non-uniform sampling and stable reconstruction problem for that prototypical space will be discussed in the subsequent paper [58]. Our prototypical space Vq (8, 3), for modeling time signals with finite rate of innovation, is given by ( ) X q Vq (8, 3) := c(λ)φλ , (c(λ))λ∈3 ∈ ` (3) , (1.1) λ∈3
where 3 is a relatively separated subset of R, `q is the space of all q-summable sequences on 3, and the generator 8 := {φλ , λ ∈ 3} is enveloped by a function h in a Wiener amalgam space Wq (Lp,u ), |φλ (x)| ≤ h(x − λ) for all λ ∈ 3,
(1.2)
see Example 2.10 for details, and see also equation (2.13) for another convenient well-localization assumptionP on the generat
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