Basic Properties of Wavelets

A wavelet multiplier is a function whose product with the Fourier transform of a wavelet is the Fourier transform of a wavelet. We characterize the wavelet multipliers, as well as the scaling function multipliers and low pass lter multipliers. We then prove that, if the set of all wavelet multipliers acts on the set of all MRA wavelets, the orbits are the sets of all MRA wavelets whose Fourier transforms have equal absolute values, and these are also equal to the sets of all MRA wavelets with the corresponding scaling functions having same absolute values of their Fourier transforms. As an application of these techniques we prove that the set of MRA wavelets is arcwise connected in L 2 (R). Foreword Two groups of collaborators, one led by Xingde Dai and David Larson and the other led by Eugenio Hernn andez and Guido Weiss, became aware that they were involved in the study of wavelets and obtained results that had much in common, even though diierent methods were employed. Early in 1996 we decided to exchange our ideas by holding meetings attended by us, our collaborators and students and by correspondence. This paper is what we hope is the rst of a series of reports describing our joint results. Fourteen researchers are involved in this work. A few shorter notes describe 1 more technical aspects of the work performed by diierent subgroups of this collection of fourteen researchers. At the end of this article, we shall describe the group that we call the \WUTAM CONSORTIUM". This is not the name of a Chinese city; it is the acronym for Washington University, Texas A & M, the two institutions associated with each of us.