Dirichlet Series and Convolution Equations

Abstract

In this paper we wish to propose a totally new approach to the theory of Dirichlet series, with the hope that our method will be able to shed new light on some old problems in the area. Our viewpoint is to consider Dirichlet series as a very particular case of solutions of homogeneous convolution equations, and to consider these objects in the framework of a general theory of Fourier analysis on non-algebraic varieties. In view of the character of this paper (which has to be considered preliminary, even though a relatively large number of results is obtained), we would like to briefly describe the philosophy which underlies our approach. From a historical point of view, the theory of Fourier integrals has been devised to be able to represent an arbitrary function as a linear combination of exponentials. Thus, functions on R (or Rn) which are sufficiently small at infinity, were given such representations with the use of exponentials with real frequencies. When larger growths at infinity had to be handled, it became necessary to allow the use of exponentials with complex frequencies and, as an immediate consequence, the Fourier integral representations became essentially non-unique, as every exponential exp(jc-z), z^C, has many integral representations (via the Cauchy formula) as a limit of linear combinations of exponentials. In the sixties, these natural considerations led L. Ehrenpreis to consider Fourier integral representations of (for example) C°° functions, with the specific purpose of finding out the existence of subsets S of C (which Ehrenpreis called sufficient sets) such that integral representations could exist with frequencies in S. Ehrenpreis developed a quite powerful theory (the so called theory of ^4t/-spaces [7]) with the idea of studying integral representations with exponentials whose frequencies were restricted to belong to some algebraic variety in C (or Cn). This study, in particular, provided a wealth of results on solutions of systems of linear constant coefficients partial differential equations (in which case S is the