Mean-periodic function

Definition

Consider a continuous complex-valued function f of a real variable. The function f is periodic with period a precisely if for all real x, we have . This can be written as

: \int f(x-t) , d\mu(t) = 0\qquad\qquad(1)

where \mu is the difference between the Dirac measures at 0 and a. The function f is mean-periodic if it satisfies the same equation (1), but where \mu is some arbitrary nonzero measure with compact (hence bounded) support.

Equation (1) can be interpreted as a convolution, so that a mean-periodic function is a function f for which there exists a compactly supported (signed) Borel measure \mu for which f*\mu = 0.

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 sense of Bohr). 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(at) where n is any non-negative integer and a is any complex number; also Df is a mean periodic function (ie mean periodic) and if h is an exponential polynomial, then the pointwise product of f and h is mean periodic).

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(D) is a linear differential operator with constant co-efficients, and L(D)f = g, then f is mean periodic if and only if g is mean periodic.

For linear differential difference equations such as Df(t) - af(t - b) = 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 (functions related to) 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.

References

References

1. (1935). "Les fonctions moyenne-périodiques". Journal de Mathématiques Pures et Appliquées.
2. (1959). "Lectures on Mean Periodic Functions". Tata Institute of Fundamental Research, Bombay.
3. (1954). "Fonctions moyenne-périodiques (d'après J.-P. Kahane)". Séminaire Bourbaki.
4. (1947). "Théorie générale des fonctions moyenne-périodiques". Ann. of Math..
5. Laird, P. G.. (1972). "Some properties of mean periodic functions". Journal of the Australian Mathematical Society.
6. (2012). "Mean-periodicity and zeta functions". Annales de l'Institut Fourier.