Mean-periodic function


In mathematical analysis, the concept of a mean-periodic function is a generalization introduced in 1935 by Jean Delsarte of the concept of a periodic function. Further results were made by Laurent Schwartz and J-P Kahane.

Definition

Consider a continuous complex-valued function of a real variable. The function is periodic with period precisely if for all real, we have. This can be written as
where is the difference between the Dirac measures at 0 and a. The function is mean-periodic if it satisfies the same equation, but where is some arbitrary nonzero measure with compact support.
Equation can be interpreted as a convolution, so that a mean-periodic function is a function for which there exists a compactly supported Borel measure for which.
There are several well-known equivalent definitions.

Relation to almost periodic functions

Mean-periodic functions are a separate generalization of periodic functions from the almost periodic functions. For instance, exponential functions are mean-periodic since, but they are not almost periodic as they are unbounded. Still, there is a theorem which states that any uniformly continuous bounded mean-periodic function is almost periodic. In the other direction, there exist almost periodic functions which are not mean-periodic.

Some basic properties

If f is a mean periodic function, then it is the limit of a certain sequence of exponential polynomials which are finite linear combinations of term t^^n exp where n is any non-negative integer and a is any complex number; also Df is a mean periodic function.
If f and g are mean periodic then f + g and the truncated convolution product of f and g is mean periodic. However, the pointwise product of f and g need not be mean periodic.
If L is a linear differential operator with constant co-efficients, and Lf = g, then f is mean periodic if and only if g is mean periodic.
For linear differential difference equations such as Df - af = g where a is any complex number and b is a positive real number, then f is mean periodic if and only if g is mean periodic.

Applications

In work related to the Langlands correspondence, the mean-periodicity of certain zeta functions associated to an arithmetic scheme have been suggested to correspond to automorphicity of the related L-function. There is a certain class of mean-periodic functions arising from number theory.