Mean periodic function

In mathematical analysis, the concept of a mean-periodic function is a generalization introduced by Jean Delsarte, of the concept of a periodic function.[1]

Consider a complex-valued function ƒ of a real variable. The function ƒ is periodic with period a precisely if for all real x, we have ƒ(x) − ƒ(x − a) = 0. This can be written as

where μ is the difference between the Dirac measures at 0 and a. A mean-periodic function is a function ƒ satisfying (1) for some nonzero measure μ with compact (hence bounded) support.