A Distributional Approach to Generalized Stochastic Processes on Locally Compact Abelian Groups

This paper is dedicated to Paul Butzer on the occasion of his 85th birthday. His work and example have strongly influenced not only the first author, but also generations of mathematicians working in approximation theory and Fourier analysis. He has shown younger colleagues the importance of remaining open to applied areas, avoiding an overly narrow scope, and exploring different ways of understanding mathematical facts. A recurring theme in his work is the logical equivalence of fundamental statements in analysis.
It may be less widely known that, besides his central role in approximation theory, Paul Butzer has also made significant contributions to probability theory. We hope that he will enjoy this note, which shows that a purely functional-analytic treatment of generalized stochastic processes is possible. The approach is based on the Segal algebra S0(G) and avoids several technical difficulties associated with the customary framework of vector-valued integration and topological vector spaces.