AN ALGEBRAIC APPROACH TO CONVOLUTIONS AND TRANSFORM METHODS

We show that the Hopf algebra dual of the polynomials in one variable appears often in analysis, but under different disguises that include proper rational functions, exponential polynomials, shift invariant op erators, Taylor functionals, and linearly recurrent sequences. The iso morphisms from the proper rational functions to the other algebras yie ld an explicit and general method for the solution of linear functiona l equations, which can be considered as an algebraic version of the us ual integral transform methods. We also explain how some of the usual convolution products in spaces of functions arise in an algebraic way. (C) 1997 Academic Press.