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Decomposition of Hardy Functions into Square Integrable Wavelets of Constant Shape
Abstract
An arbitrary square integrable real-valued function (or, equivalently, the associated Hardy function) can be conveniently analyzed into a suitable family of square integrable wavelets of constant shape, (i.e. obtained by shifts and dilations from any one of them.) The resulting integral transform is isometric and self-reciprocal if the wavelets satisfy an “admissibility condition” given here. Explicit expressions are obtained in the case of a particular analyzing family that plays a role analogous to that of coherent states (Gabor wavelets) in the usual $L_2 $ -theory. They are written in terms of a modified $\Gamma $-function that is introduced and studied. From the point of view of group theory, this paper is concerned with square integrable coefficients of an irreducible representation of the nonunimodular $ax + b$-group.
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Metadata
- Authors
- Grossmann, A., & Morlet, J.
- Venue
- SIAM Journal on Mathematical Analysis
- Published
- 1984-07
- Volume
- 15
- Issue
- 4
- Pages
- 723-736
- ISSN
- 0036-1410 (ISSN-L)
- 1095-7154
- OpenAlex
- W2096684483
- Cited by
- 3,624
- References
- 0
- FWCI
- 3.57
- Type
- journal article
- Language
- en
- Open access
- closed
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- Grossmann, A., & Morlet, J. (1984). Decomposition of Hardy Functions into Square Integrable Wavelets of Constant Shape. SIAM Journal on Mathematical Analysis. https://doi.org/10.1137/0515056
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