Decomposition of Hardy functions into square integrable wavelets of constant shape
A. Grossmann, J. Morlet · Princeton University Press eBooks · 2009
function)canbeconvenientlyanalyzedintoasuitablefamilyof squareintegrablewaveletsof constantshape, (i.e.obtainedbyshiftsanddilationsfromanyoneof them.)Theresultingintegraltransformis isometricand self-reciprocalif thewaveletsatisfyan "admissibilitycondition"givenhere.Explicitexpressionsareob-tainedin thecaseof a particularanalyzingfamilythatplaysa roleanalogousto thatof coherentstates (Gabor wavelets)in the usualLz-theory.They arewrittenin termsof a modifiedf-function thatis introducedand studied.From thepoint of viewof grouptheory,thispaperis concernedwith square integrablecoefficientsof anirreduciblerepresentationf thenonunimodularx+b-group. 1. Introduction. 1.1. It is wellknownthatan arbitrarycomplex-valuedsquareintegrablefunction l/;(t) admitsa representationby Gaussians,shiftedin directandFouriertransformed space.If g(t) =2- 1/271'- 3/4e- t2/2 andto ' Wo arearbitraryreal,consider (1.1) g(to,wo)(t) =e-iwoto/2eiwotg(t- to)