Decomposition of Hardy functions into square integrable wavelets of constant shape. (English) Zbl 0578.42007

Grossmann, A.; Morlet, J.

SIAM J. Math. Anal. 15, 723-736 (1984).

Summary: 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.

MSC:

42A38	Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
42C40	Nontrigonometric harmonic analysis involving wavelets and other special systems

Keywords:

square integrable real-valued function; Hardy function; square integrable wavelets of constant shape; Gabor wavelets; square integrable coefficients; irreducible representation