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Fourier-like frames on locally compact abelian groups. (English) Zbl 1311.42075

Several parts of classical frame theory for structured function systems (e.g. Gabor systems and generalized shift-invariant systems on \(R^s\)) have parallel versions on locally compact abelian (LCA) groups. For example, the Ron and Shen theory for shift-invariant spaces was generalized to this setting by C. Cabrelli and V. Paternostro [J. Funct. Anal. 258, No. 6, 2034–2059 (2010; Zbl 1190.43003)], while the equations characterizing Parseval frames via generalized shift-invariant systems were obtained in the LCA group setting by G. Kutyniok and D. Labate [Colloq. Math. 106, No. 2, 197–220 (2006; Zbl 1113.43008)]. The paper under review considers a class of functions, called Fourier-like systems, defined on an LCA group by letting a class of modulation operators act on a given countable collection of functions. Via the Fourier transform these systems correspond to generalized shift-invariant systems. The focus is on simple sufficient conditions and explicit constructions, rather than on characterizations. In particular, the authors derive sufficient conditions for such a class of functions to form a Bessel sequence or a frame and for two such systems to be dual frames. The constructions are based on a generalization of the classical B-splines to the setting of LCA groups that were obtained independently by S. Dahlke [in: Wavelets, images, and surface fitting. Papers from the 2nd international conference on curves and surfaces, held in Chamonix-Mont-Blanc, France, 1993. Wellesley, MA: A K Peters. 141–156 (1994; Zbl 0834.43004)] and V. M. Tikhomirov [Russ. Math. Surv. 49, No. 3, 200–201 (1994); translation from Usp. Mat. Nauk 49, No. 3(297), 193–194 (1994; Zbl 0848.41010)]. These constructions show that the theory is applicable in practice on LCA groups, not just on the formal level of deriving the theorems.
The paper is organized as follows. Section 1 is introductory. Section 2 contains a brief introduction to classical harmonic analysis on LCA groups, with focus on the relationship between the group and its dual group, and the Fourier transform. Section 3 contains the main results and the explicit frame constructions, while Section 4 translates the results into the setting of generalized shift-invariant systems.

MSC:

42C15 General harmonic expansions, frames
22B05 General properties and structure of LCA groups
43A85 Harmonic analysis on homogeneous spaces
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