Abstract
The Zak transform has been used in engineering and applied mathematics for several years and many purposes. In this paper, we show how it can be used to obtain an exceedingly elementary proof of the Plancherel theorem and for developing many results in Harmonic Analysis in particularly direct and simple ways. Many publications state that it was introduced in the middle sixties. It is remarkable that only a small number of mathematicians know this and that many textbooks continue to give much harder and less transparent proofs of these facts. We cite a 1950 paper by I. Gelfand and a book by A. Weil, written in 1940 that indicate that in a general non-compact LCA setting the Fourier transform is an average of Zak transforms (which are really Fourier series expressions). We actually introduce versions of these transforms that show how naturally and simply one obtains these results.
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References
I. Gelfand, Eigenfunction expansions for an equation with periodic coefficients, Dokl. Akad. Nauk. SSSR, 76 (1950), pp. 1117-1120 (in Russian).
K. Gröchenig, Aspects of Gabor analysis on locally compact abelian groups, in: Gabor Analysis and Algorithms, H. G. Feichtinger and E. Strohmer, Editors. Birkhäuser, (2001), pp. 211-229.
E. Hernández, H. Šikić, G. Weiss, E. N. Wilson, On the properties of the integer translates of a square integrable function, Contemp. Math. 505, (2010), pp. 233-249.
E. Hernández, H. Šikić, G. Weiss, E. N. Wilson, Cyclic subspaces for unitary representations of LCA groups; generalized Zak transforms, to appear in a volume dedicated to Andrzej Hulanicki publ. by Coll. Math in February (2010).
A. J. E. M. Janssen, The Zak transform: a signal transform for sampled time-continuous signals, Philips J. Research, 43 (1988), pp. 23-69.
A. Weil, L’integration dans les groupes topologiques et ses applications, Hermann (Paris) (1951).
J. Zak, Finite translations in solid state physics, Phys. Rev. Lett. 19 (1967), pp. 1385-1397.
Acknowledgements
The research of E. Hernández is supported by grants MTM2007 − 60952 of Spain and SIMUMAT S-0505/ESP-0158 of the Madrid Community Region. The research of H. Šikić, G. Weiss and E. Wilson is supported by the US-Croatian grant NSF-INT-0245238. The research of H. Šikić is also supported by the MZOS grant \(037 - 0372790 - 2799\) of the Republic of Croatia.
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Hernández, E., Šikić, H., Weiss, G.L., Wilson, E.N. (2011). The Zak Transform(s). In: Cohen, J., Zayed, A. (eds) Wavelets and Multiscale Analysis. Applied and Numerical Harmonic Analysis. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-8095-4_8
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DOI: https://doi.org/10.1007/978-0-8176-8095-4_8
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