Parabolic wavelet transforms associated with the singular heat operators $-{\Delta }_{\gamma }+\partial /\partial t$ and $I-{\Delta }_{\gamma }+\partial /\partial t$, where ${\Delta }_{\gamma }={\sum }_{k=1}^n\partial ²/\partial x{²}_k+(2\gamma /xₙ)\partial /\partial xₙ$, are introduced. These transforms are defined in terms of the relevant generalized translation operator. An analogue of the Calderón reproducing formula is established. New inversion formulas are obtained for generalized parabolic potentials representing negative powers of the singular heat operators.

In this paper we shall compare three notions of pointwise smoothness: the usual definition, J.M. Bony's two-microlocal spaces Cx0s,s', and the corresponding definition on the wavelet coefficients. The purpose is mainly to show that these two-microlocal spaces provide "good substitutes" for the pointwise Hölder regularity condition; they can be very precisely compared with this condition, they have more functional properties, and can be characterized by conditions on the wavelet coefficients. We...

We show that (bi-orthogonal) wavelet bases associated to a dilation matrix which is compatible with integer shifts are generally provided by a multi-resolution analysis. The proof is done by studying the projectors which commute with integer shifts.

We give some new properties of refinable measures and survey results on their asymptotic normality. We also give a survey on the asymptotically optimal time-frequency localisation of refinable measures and associated wavelets.