Sequence Dominance in Shift-Invariant Spaces
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(2020) 26:55
Sequence Dominance in Shift-Invariant Spaces Tomislav Beri´c1 · Hrvoje Šiki´c1 Received: 3 January 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We show that a Bessel sequence Bψ of integer translates of a square integrable function ψ ∈ L 2 (R) has the Besselian property if and only if its periodization function pψ is bounded from below. We also give characterizations of Besselian and Hilbertian properties of a general sequence Bψ of integer translates in terms of the classical notion of sequence dominance. Keywords Shift invariant systems · Bases · Frames · Riesz bases · Periodization function · Besselian property · Hilbertian property Mathematics Subject Classification Primary 42C15 · Secondary 42A20
1 Introduction Reproducing function systems, like wavelets, Gabor systems, shearlets etc., have attracted a lot of attention from theoretical aspect and from the applications point of view. The role of shift invariant spaces is exceptionally important in the theory of such systems; see [2,6,18,19]. Very soon it became clear that a wealth of information is hidden in the properties of the “main resolution level” and that such properties affect the entire system; see [5,16,17,22], as early examples. This naturally directed us toward the study of principal shift invariant spaces ψ := span {Tk ψ := ψ(· − k) : k ∈ Z} ,
Communicated by Chris Heil.
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Tomislav Beri´c [email protected] Hrvoje Šiki´c [email protected]
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Department of Mathematics, University of Zagreb, Bijeniˇcka cesta 30, 10000 Zagreb, Croatia 0123456789().: V,-vol
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Journal of Fourier Analysis and Applications
(2020) 26:55
where ψ ∈ L 2 (R); see [15] and [8] for early results. The approach to study wavelets from this point of view enabled a rather comprehensive theory, developed most recently in [13]. This paper is largely influenced by [8] and [13] (see Chapter I, in particular). Despite a possibility to generalize many results even to the level of LCA groups (see [7]), some questions remained open even in the most basic case of functions in L 2 (R). In order to treat these questions, let us remind our readers that many properties have already been developed for bases in Banach spaces decades ago (see [23] and [12] as good sources for the basis theory). Such properties play the role beyond bases in Banach spaces, for more general systems like frames, minimal systems, “infinite sums-linear independence”, etc. In particular, the notions of Besselian and Hilbertian Schauder bases were extended and studied for systems of translates in [8] (and, again, in [13]). An interesting dichotomy develops here, where the “Hilbertian” properties are relatively easy to handle, but “Besselian” ones (despite their “dual” nature) require much more effort. In this paper we deal with several open questions. We provide significant new theorems that give answers in special cases (but quite general, nevertheless). The key question (see (3.10) in this paper) on the characterization of the “Besselian” cond
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