Mean-periodic function

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

Definition

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

${\displaystyle \int f(x-y)\,d\mu (y)=0\qquad \qquad (1)}$

where ${\displaystyle \mu }$ 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 ${\displaystyle \mu }$ 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 ${\displaystyle \mu }$ for which ${\displaystyle f*\mu =0}$.[4]

There are several well-known equivalent definitions.[2]

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 exp(x+1) − e.exp(x) = 0, 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.[2]

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

See also

• 　almost periodic functions

References

1. Delsarte, Jean (1935). "Les fonctions moyenne-périodiques". Journal de Mathématiques Pures et Appliquées. 17: 403–453.
2. Kahane, J.-P. (1959). Lectures on Mean Periodic Functions (PDF). Tata Institute of Fundamental Research, Bombay.
3. Malgrange, Bernard (1954). "Fonctions moyenne-périodiques (d'après J.-P. Kahane)" (PDF). Séminaire Bourbaki (97): 425–437.
4. Schwartz, Laurent (1947). "Théorie générale des fonctions moyenne-périodiques" (PDF). Ann. of Math. 48 (2): 857–929. doi:10.2307/1969386. JSTOR 1969386.
5. Fesenko, I.; Ricotta, G.; Suzuki, M. (2012). "Mean-periodicity and zeta functions". Annales de l'Institut Fourier. 62 (5): 1819–1887. arXiv:0803.2821. doi:10.5802/aif.2737.