Weyl-Heisenberg frames and Riesz bases in \(L_2(\mathbb{R}^d)\). (English) Zbl 0892.42017

From the introduction: “The present paper is the second in a series of three, all devoted to the study of shift-invariant frames and shift-invariant stable (= Riesz) bases for \(H:= L_2(\mathbb{R}^d)\), \(d\geq 1\), or a subspace of it. In the first paper [RS1] [Can. J. Math. 47, No. 5, 1051-1094 (1995; Zbl 0838.42016)], we studied such bases under the mere assumption that the basis set can be written as a collection of shifts (namely, integer translates) of a set of generators \(\Phi\). The present paper analyzes Weyl-Heisenberg (WH, known also as Gaborian) frames and stable bases. Aside from specializing the general methods and results of [RS1] to his important case, we exploit here the special structure of the WH set, and in particular the duality between the shift operator and the modulation operator, the latter being absent in the context of general shift-invariant sets. In the third paper [RS3] [J. Funct. Anal. 148, No. 2, 408-447 (1997; Zbl 0891.42018)], we present applications of the results of [RS1] to wavelet (or affine) frames. The flavour of the results there is quite different; wavelet sets are not shift-invariant, and the main effort of [RS3] is to show that, nevertheless, the basic analysis of [RS1] does apply to that case as well”.


42C15 General harmonic expansions, frames
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