Wavelets and Multiscale Analysis pp 151-157 | Cite as
The Zak Transform(s)
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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Notes
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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