Parabolic-Like Wavelet Transforms and Relevant Reproducing Formulas

Abstract

We introduce new anisotropic wavelet-type transforms generated by two components: a wavelet measure (or a wavelet function) and a kernel function that naturally generalizes the Gauss and Poisson kernels. The analogues of Calderon’s reproducing formula are established in the framework of the $$L_{p}(\mathbb {R}^{n+1})$$ -theory. These wavelet-type transforms have close connection with a significant generalization of the classical parabolic-Riesz and parabolic-Bessel potentials and can be used to find explicit inversion formulas for the generalized parabolic-type potentials.