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On the pseudodilation representations of Flornes, Grossmann, Holschneider, and Torrésani. (English) Zbl 0904.42023

In [J. Comput. Harmon. Anal. 1, No. 2, 137-146 (1994; Zbl 0798.42021)], K. Flornes, A. Grossmann, M. Holschneider and B. Torrésani considered continuous wavelet transforms for sequences of length \(p\) over finite fields \({\mathbb Z}_p\), \(p\) prime. The usual dilation operator does not allow to be interpreted as sampling of dilated functions in \(L^2({\mathbb R})\). Therefore pseudodilations are introduced, which give a smoother wavelet transform. In this paper these pseudodilations are further investigated for general finite fields \(F\). It is shown that the class of compatible filters is completely parametrized by the torus \(T^{p^k-1}\). Also the wavelet transforms for complex valued functions in \(\ell^2(F)\) associated with these pseudodilations are discussed. It is shown that the energy conservation law holds, so that unitary wavelet transforms are possible. Some examples are given.

MSC:

42C15 General harmonic expansions, frames

Citations:

Zbl 0798.42021
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References:

[1] Flornes, K., Grossmann, A., Holschneider, M., and Torrésani, B. (1994). Wavelets on discrete fields.Appl. Computational Harmonic Analysis. 1, (2), 137–146. · Zbl 0798.42021 · doi:10.1006/acha.1994.1001
[2] Grossmann, A., Morlet, J., and Paul, T. (1985). Transforms associated to square-integrable group representations, I, General results.J. Math. Phys., 26, 2473–2479. · Zbl 0571.22021 · doi:10.1063/1.526761
[3] Hilton, P., and Stammbach, U. (1971). A Course in Homological Algebra. Graduate Texts in Mathematics series No. 4, Springer-Verlag, Berlin. · Zbl 0238.18006
[4] Lidl, R. and Pilz, G. (1984). Applied Abstract Algebra. Undergraduate Texts in Mathematics Series, Springer-Verlag, Berlin. · Zbl 0572.00001
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