Mean-periodic function explained

In mathematical analysis, the concept of a mean-periodic function is a generalization introduced in 1935 by Jean Delsarte^{[1] [2]} of the concept of a periodic function. Further results were made by Laurent Schwartz and J-P Kahane.^{[3] [4]}

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

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

where

\mu is the difference between the Dirac measures at 0 and a. The function 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 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.^[5]

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.^[6] There is a certain class of mean-periodic functions arising from number theory.

See also

• almost periodic functions

Notes and References

1. Jean . Delsarte . Les fonctions moyenne-périodiques . Journal de Mathématiques Pures et Appliquées . 17 . 1935 . 403–453.
2. Book: J.-P. . Kahane . Jean-Pierre Kahane . Lectures on Mean Periodic Functions . 1959 . Tata Institute of Fundamental Research, Bombay .
3. Bernard . Malgrange . Bernard Malgrange. 1954 . Fonctions moyenne-périodiques (d'après J.-P. Kahane). Séminaire Bourbaki. 97. 425–437.
4. Laurent. Schwartz . Laurent Schwartz . Théorie générale des fonctions moyenne-périodiques . Ann. of Math. . 1947 . 857–929 . 48 . 2. 10.2307/1969386 . 1969386 .
5. Laird . P. G. . 1972 . Some properties of mean periodic functions . Journal of the Australian Mathematical Society . 14 . 4 . 424–432 . 10.1017/s1446788700011058 . 0004-9735.
6. M.. Suzuki. G.. Ricotta. I.. Fesenko. Ivan Fesenko. Mean-periodicity and zeta functions. Annales de l'Institut Fourier. 2012. 62. 1819–1887. 5. 0803.2821. 10.5802/aif.2737.