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 anyone 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 Lz-theory. They are written in terms of a modified f-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. 1.2. Consider now the case where the object of interest is not a complex-valued function~(t), but a square integrable real-valued function s(t), say the wiggle of a seismograph. It has been known for a long time that it is very useful to consider set) as the real part of a complex-valued square integrable function h(t) which has the special property that its Fourier transform vanishes on a half-line (say h(w)=O for w<O).

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