On a Group of Unitary Deformations

Abstract

We develop in this work an idea initially due to J.M.Levy-Leblond ([10]) for deforming through explicit homotopies, the Fourier transform into identity, the convolution product into pointwise product, differential operators into others, et cetera, and we express here our best thanks to him. The main tools to achieve this are the Hermite expansion of any tempered distribution, due to L.Schwartz ([6]) but not much used since, and Mahler’s formula on Hermite functions, which allows us an explicit computation of the kernels of all the operators involved.