A unified characterization of reproducing systems generated by a finite family, II

Abstract

By a "reproducing" method for H = L 2 (R n) we mean the use of two countable families {e α : α ∈ A}, {f α : α ∈ A}, in H, so that the first "analyzes" a function h ∈ H by forming the inner products {< h, e α >: α ∈ A}, and the second "reconstructs" h from this information: h = α∈A < h, e α > f α. A variety of such systems have been used successfully in both pure and applied mathematics. They have the following feature in common: they are generated by a single or a finite collection of functions by applying to the generators two countable families of operators that consist of two of the following three actions: dilations, modulations, and translations. The Gabor systems, for example, involve a countable collection of modulations and translations; the affine systems (that produce a variety of wavelets) involve translations and dilations. A considerable amount of research has been conducted in order to characterize those generators of such systems. In this article we establish a result that "unifies" all of these characterizations by means of a relatively simple system of equalities. Such unification has been presented in a work by one of the authors. One of the novelties here is the use of a different approach that provides us with a considerably more general class of such reproducing systems; for example, in the affine case, we need not to restrict the dilation matrices to ones that preserve the integer lattice and are expanding on R n. Another novelty is a detailed analysis, in the case of affine and quasi-affine systems, of the characterizing equations for different kinds of dilation matrices. Math Subject Classifications. 42C15, 42C40.

The theory of reproducing systems on locally compact abelian groups

“…This result recovers Theorem 5.9 in [13] and, as shown in that paper, it generalizes and contains all classical characterization results about affine systems, including those in [4,6,11,23]. We refer to the same paper for more details about the motivation and history of these and similar characterization equations for the affine systems in L 2 (R n ).…”

“…Several papers have been devoted to the formulation and study of these characterizations, and they play a major role in the construction and study of Gabor and affine systems (for example, [3,4,6,10,11,14,17,19,[21][22][23]). The approach that we develop in this paper adapts some ideas from [13,19], where one of the present authors has developed a unified approach to Gabor systems and affine systems in L 2 (R n ).…”

“…The D-bracket product of f and g, which was originally introduced in [2] and extended in [13], is defined, in our setting, as (10) […”

History and Evolution of the Density Theorem for Gabor Frames

“…A comparison of these various definitions is made is [137]. In [139] it is shown that the local integrability condition of [114] is equivalent to finite affine Beurling density (assuming some mild regularity on the wavelet ψ). The role of density conditions in wave packet systems (which combine translations, modulations, and dilations together) is explored in [47].…”

Operators commuting with a discrete subgroup of translations

“…Using the distributional version Theorem 1.2 makes the derivation of these important results conceptually simple and in addition avoids the technicalities connected with the L 2 -boundedness. For a unified treatment of reproducing systems, see [19,28,30].…”