Frame properties of wave packet systems in $L^2({\mathbb R}^d)$
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Frame properties of wave packet systems in L2 (R d ) Ole Christensen · Asghar Rahimi
Received: 16 May 2006 / Accepted: 11 December 2006 / Published online: 8 May 2007 © Springer Science + Business Media B.V. 2007
Abstract Extending work by Hernandez, Labate and Weiss, we present a sufficent condition for a generalized shift-invariant system to be a Bessel sequence or even a frame for L2 (Rd ). In particular, this leads to a sufficient condition for a wave packet system to form a frame. On the other hand, we show that certain natural conditions on the parameters of such a system exclude the frame property. Keywords Frames · Wave packet systems · Generalized shift-invariant systems Mathematics Subject Classifications (2000) 42C15 · 42C40
1 Introduction In this paper we consider frame properties for systems of functions generated by combined action of dilation, translation and modulation on a function in L2 (Rd ). At the first time, systems of this form were used in [4] by Cordoba and Fefferman in the study of some classes of singular integral operators. They called these systems wave packet systems. Wave packet systems have been considered and extended by several authors, see [3, 5, 7–9, 11]. One of the most important applications appears in the work of Lacey and Thiele on boundedness of the bilinear Hilbert transform [12, 13]. In [5], Czaja, Kutyniok and Speegle proved that certain geometric conditions on the
Communicated by Juan Manuel Peña. O. Christensen (B) Department of Mathematics, Technical University of Denmark, Building 303, 2800 Lyngby, Denmark e-mail: [email protected] A. Rahimi Department of Mathematics, University of Maragheh, Maragheh, Iran e-mail: [email protected]
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O. Christensen, A. Rahimi
set of parameters in a wave packet systems are necessary in order for the system to form a frame. In the present paper we consider wave packet systems as special cases of generalized shift-invariant systems, a concept studied by Hernandez, Labate and Weiss in [8], Ron and Shen in [16], as well as Eldar and the second-named author in [2]. Extending a result in [11], we present a sufficent condition for a generalized shiftinvariant system to be a Bessel sequence or even a frame for L2 (Rd ); in particular this leads to a sufficient condition for a wave packet system to form a frame. Based on a result from [8], we also present some negative results; in fact, certain natural conditions on the parameters in a wave packet system exclude the frame property. The main results appear in Section 3. Section 2 contains some basic definitions and results.
2 Preliminaries For y ∈ Rd , the translation operator T y acting on f ∈ L2 (Rd ) is defined by (T y f )(x) = f (x − y), x ∈ Rd . For y ∈ Rd , the modulation operator E y is (E y f )(x) = e2πiy·x f (x), x ∈ Rd , where y · x denotes the inner product between y and x in Rd . The dilation operator associated with a real d × d matrix C is (DC f )(x) = | det C|1/2 f (Cx), x ∈ Rd . We are now ready to define the main concepts considered in this paper. We begin
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