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.
Keywords
Square-integrable functionMathematicsSquare (algebra)WaveletIntegrable systemConstant (computer programming)Mathematical analysisPure mathematicsReciprocalFunction (biology)Geometry
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Publication Info
- Year
- 1984
- Type
- article
- Volume
- 15
- Issue
- 4
- Pages
- 723-736
- Citations
- 3462
- Access
- Closed
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Cite This
A. Großmann,
J. Morlet
(1984).
Decomposition of Hardy Functions into Square Integrable Wavelets of Constant Shape.
SIAM Journal on Mathematical Analysis
, 15
(4)
, 723-736.
https://doi.org/10.1137/0515056
Identifiers
- DOI
- 10.1137/0515056