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Wavelets

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Coherent States, Wavelets, and Their Generalizations

Part of the book series: Theoretical and Mathematical Physics ((TMP))

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Abstract

This is the first of four chapters devoted to a very successful type of CS, namely, wavelets. In the present one, we treat the simplest case, the continuous wavelet transform (CWT) in 1-D. Starting from the beginning, we rewrite the general CS formalism for the case at hand, that is, the connected affine group of the line. We discuss the basic properties, the interpretation of the CWT as a phase space representation and some examples, with emphasis on a recent application to NMR spectroscopy.

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Notes

  1. 1.

    Sometimes called the Gabor transform, although Gabor considered only a discretized version of it [294].

  2. 2.

    The set {α m } is directed if, for any pair α m , α n , there is an element α p such that α m ≤ α p and α m ≤ α p .

  3. 3.

    By this, we mean that the function is numerically negligible outside that region.

  4. 4.

    We return here to the (t, ω) notations, more familiar in signal processing.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Concordia University, Montréal, Québec, Canada

    Syed Twareque Ali

  2. Institut de Recherche en Mathématique et Physique (IRMP), Université Catholique de Louvain, Louvain-la-Neuve, Belgium

    Jean-Pierre Antoine

  3. Astroparticules et Cosmologie (APC, UMR 7164), Université Paris Diderot, Sorbonne Paris Cité Paris, France

    Jean-Pierre Gazeau

  4. Centro Brasileiro de Pesquisas Fisicas (CBPF), Rio de Janeiro, Brasil

    Jean-Pierre Gazeau

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Ali, S.T., Antoine, JP., Gazeau, JP. (2014). Wavelets. In: Coherent States, Wavelets, and Their Generalizations. Theoretical and Mathematical Physics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-8535-3_12

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