An Approach to the Study of Wave Packet Systems

The traditional study of reproducing systems involves the Gabor systems, that are generated by the action of translations and modulations on a single or finite family of functions in L2(Rn), and the affine systems, where the dilations are used rather than the modulations. In this paper, we show that the interplay of all three operators yield a wide variate of reproducing systems, and we employ the term wave packet systems, which has been used by other authors, to describe those function systems generated by the combined action of a class of translations, modulations and dilations on a finite family of functions. We will examine in detail both the continuous and discrete versions of these systems. We shall show that these systems can be studied by using a unified approach that the authors have developed in some of their previous work. 1. Preliminaries Before embarking in this study, it will be useful to introduce some notation and definitions. We consider three fundamental operators on L(R): the translations Ty : (Ty f)(x) = f(x − y), where y ∈ R; the dilations Da : (Da f)(x) = |det a|1/2f(ax), where a ∈ GLn(R); and the modulations Mν : (Mν f)(x) = e2πiν·xf(x), where ν ∈ R. The Fourier transform is defined as f̂(ξ) = (Ff)(ξ) = ∫ Rn f(x) e−2πiξ·x dx, and the inverse Fourier transform is f̌(x) = (F−1f)(x) = ∫