International Journal of Mathematics and Mathematical Sciences
Volume 3 (1980), Issue 2, Pages 199-235
doi:10.1155/S0161171280000154

Mean-periodic functions

Carlos A. Berenstein1 and B. A. Taylor2

1Department of Mathematics, University of Maryland, College Park 20742, Maryland, USA
2Department of Mathematics, University of Michigan, Ann Arbor 48109, Michigan, USA

Received 25 September 1979

Copyright © 1980 Carlos A. Berenstein and B. A. Taylor. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We show that any mean-periodic function f can be represented in terms of exponential-polynomial solutions of the same convolution equation f satisfies, i.e., u∗f=0(μ∈E′(ℝn)). This extends to n-variables the work of L. Schwartz on mean-periodicity and also extends L. Ehrenpreis' work on partial differential equations with constant coefficients to arbitrary convolutors. We also answer a number of open questions about mean-periodic functions of one variable. The basic ingredient is our work on interpolation by entire functions in one and several complex variables.