Christensen, Ole; Goh, Say Song Fourier-like frames on locally compact abelian groups. (English) Zbl 1311.42075 J. Approx. Theory 192, 82-101 (2015). 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. Reviewer: Richard A. Zalik (Auburn) Cited in 18 Documents MSC: 42C15 General harmonic expansions, frames 22B05 General properties and structure of LCA groups 43A85 Harmonic analysis on homogeneous spaces Keywords:frames; dual frames; Bessel sequences; locally compact abelian groups; generalized shift-invariant spaces; modulation operators; Fourier transform Citations:Zbl 1190.43003; Zbl 1113.43008; Zbl 0834.43004; Zbl 0848.41010 PDFBibTeX XMLCite \textit{O. Christensen} and \textit{S. S. Goh}, J. Approx. Theory 192, 82--101 (2015; Zbl 1311.42075) Full Text: DOI References: [1] Cabrelli, C.; Paternostro, V., Shift-invariant spaces on LCA groups, J. Funct. Anal., 258, 2034-2059 (2010) · Zbl 1190.43003 [2] Cariolaro, G., Unified Signal Theory (2011), Springer · Zbl 1226.94006 [3] Christensen, O., Frames and Bases. An Introductory Course (2007), Birkhäuser [4] Christensen, O.; Goh, S. S., Pairs of dual periodic frames, Appl. Comput. Harmon. Anal., 33, 315-329 (2012) · Zbl 1266.42073 [5] Christensen, O.; Massopust, P., Exponential B-splines and the partition of unity property, Adv. Comput. Math., 37, 301-318 (2012) · Zbl 1260.42021 [6] Christensen, O.; Rahimi, A., Frame properties of wave packet systems in \(L^2(R^d)\), Adv. Comput. Math., 29, 101-111 (2008) · Zbl 1152.42013 [7] Dahlke, S., Multiresolution analysis and wavelets on locally compact abelian groups, (Laurent, P.-J.; Le Méhauté, A.; Schumaker, L., Wavelets, Images, and Surface Fittings (1994), AK Peters), 141-156 · Zbl 0834.43004 [8] Feichtinger, H. G.; Kozek, W., Quantization of TF lattice-invariant operators on elementary LCA groups, (Feichtinger, H. G.; Strohmer, T., Gabor Analysis and Algorithms: Theory and Applications (1998), Birkhäuser), 233-266 · Zbl 0890.42012 [9] Gröchenig, K., Aspects of Gabor analysis on locally compact abelian groups, (Feichtinger, H. G.; Strohmer, T., Gabor Analysis and Algorithms: Theory and Applications (1998), Birkhäuser), 211-231 · Zbl 0890.42011 [10] Guo, K.; Labate, D.; Lim, W.; Weiss, G.; Wilson, E., Wavelets with composite dilations and their MRA-properties, Appl. Comput. Harmon. Anal., 20, 202-236 (2006) · Zbl 1086.42026 [11] Hewitt, E.; Ross, K., Abstract Harmonic Analysis, Vol. 1 and 2 (1963), Springer · Zbl 0115.10603 [12] Kaniuth, E.; Kutyniok, G., Zeroes of the Zak transform on locally compact abelian groups, Proc. Amer. Math. Soc., 126, 3561-3569 (1998) · Zbl 0907.43005 [13] King, E. J.; Skopina, M. A., Quincunx multiresolution analysis for \(L^2(Q_2^2)\), P-Adic Numbers Ultrametric Anal. Appl., 2, 222-231 (2010) · Zbl 1273.42036 [14] Kutyniok, G.; Labate, D., Theory of reproducing systems on locally compact abelian group, Colloq. Math., 106, 197-220 (2006) · Zbl 1113.43008 [15] Li, S., On general frame decompositions, Numer. Funct. Anal. Optim., 16, 1181-1191 (1995) · Zbl 0849.42023 [16] Reiter, H.; Stegeman, J. D., Classical Harmonic Analysis and Locally Compact Groups (2000), Oxford University Press · Zbl 0965.43001 [17] Ron, A.; Shen, Z., Generalized shift-invariant systems, Constr. Approx., 22, 1-45 (2005) · Zbl 1080.42025 [18] Rudin, W., Fourier Analysis on Groups (1962), Interscience Publishers · Zbl 0107.09603 [19] Tikhomirov, V. M., Harmonic tools for approximation and splines on locally compact abelian groups, Uspekhi Mat. Nauk, 49, 193-194 (1994), translated in Russian Math. Surveys 49 (1994), 200-201 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.