Operators on radial functions

A general theory is provided that allows to write multi-variate Fourier transforms or convolutions of radial functions as very simple univariate operations. As a byproduct, an interesting group of operators (Iα)xϵR with Iα−β = Iα Iβ = Iβ Iα is defined. It contains the classical derivatives as I1 = −d dr and is intimately connected to the Fourier transform. Applications to the construction of new positive definite radial and to new identities for special functions are included.