On some explicit deconvolution formulas

The aim is to solve the following problem: given several measuring devices defined by a convolution with distributions mu ₁, ..., mu _m of compact support in Rⁿ one would like to construct explicitly deconvolutors, i.e. distributions nu ₁, ..., nu _m, also of compact support, such that: mu ₁* nu ₁+...+ mu _m* nu _m= delta where * designs the convolution operation; such distributions nu ₁, ..., nu _m define measuring devices which, as soon as they have been constructed, allow one to reconstruct exactly an arbitrary signal phi in C^infinity (Rⁿ) which was measured through the original devices defined by convolution, with distributions mu ₁, ..., mu _m. By explicit, the authors mean that nu ₁, ..., nu _m are given by formulas involving the mu ₁, ..., mu _m, convolution, differentiation, integration and sums. The authors try to solve that kind of problem under particular hypothesis on mu ₁, ..., mu _m, so as to include all examples where the question has arisen and specially the case of the use of two optical devices whose transfer functions are 1/ pi R₁(J₁(rR₁)/r) and 1/ pi R₂(J₁(rR₂)/r) with distinct R₁ and R₂.