Decomposition of Hardy Functions into Square Integrable Wavelets of Constant Shape

TLDR

Square integrable real‑valued functions can be decomposed into a family of constant‑shape wavelets generated by shifts and dilations. The paper investigates square‑integrable coefficients of an irreducible representation of the non‑unimodular ax+b group. These coefficients are expressed via a modified Γ‑function introduced and studied in the work. The resulting integral transform is isometric and self‑reciprocal under an admissibility condition, and explicit formulas are provided for a family analogous to Gabor wavelets.

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.