Parabolic potentials and wavelet transforms with the generalized translation
Volume 145 / 2001
Studia Mathematica 145 (2001), 1-16
MSC: Primary 42C40.
DOI: 10.4064/sm145-1-1
Abstract
Parabolic wavelet transforms associated with the singular heat operators $-{\mit \Delta }_{\gamma }+{\partial /\partial t}$ and $I-{\mit \Delta }_{\gamma }+{\partial /\partial t}$, where ${\mit \Delta }_{\gamma }=\sum _{k=1}^{n} {\partial ^2/\partial x_{k}^{2}}+({2\gamma /x_{n}}) {\partial /\partial x_{n}}$, 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.
