Parabolic potentials and wavelet transforms with the generalized translation

Parabolic wavelet transforms associated with the singular heat operators -Delta gamma + partial derivative/partial derivativet and I-Delta gamma + partial derivative/partial derivativet, where Delta gamma = Sigma (n)(k=l) partial derivative (2)/partial derivativex(k)(2) + (2 gamma /x(n))partial derivative/partial derivativex(n), are introduced. These transforms are defined in terms of the relevant generalized translation operator. An analogue of the Calderon reproducing formula is established. New inversion formulas are obtained for generalized parabolic potentials representing negative powers of the singular heat operators.