Mean-periodic function
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 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
- [math]\displaystyle{ \int f(x-t) \, d\mu(t) = 0\qquad\qquad(1) }[/math]
where [math]\displaystyle{ \mu }[/math] 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 [math]\displaystyle{ \mu }[/math] 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 [math]\displaystyle{ \mu }[/math] for which [math]\displaystyle{ f*\mu = 0 }[/math].[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]
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. (Laird, P (1972), 'Some properties of mean periodic functions', J. Aust. Math. Society, 14: 424 - 432)
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
References
- ↑ "Les fonctions moyenne-périodiques". Journal de Mathématiques Pures et Appliquées 17: 403–453. 1935.
- ↑ 2.0 2.1 2.2 Lectures on Mean Periodic Functions. Tata Institute of Fundamental Research, Bombay. 1959. http://www.math.tifr.res.in/~publ/ln/tifr15.pdf.
- ↑ "Fonctions moyenne-périodiques (d'après J.-P. Kahane)". Séminaire Bourbaki (97): 425–437. 1954. http://www.numdam.org/article/SB_1951-1954__2__425_0.pdf.
- ↑ 4.0 4.1 "Théorie générale des fonctions moyenne-périodiques". Ann. of Math. 48 (2): 857–929. 1947. doi:10.2307/1969386. http://sites.mathdoc.fr/OCLS/pdf/OCLS_1947__8__857_0.pdf.
- ↑ "Mean-periodicity and zeta functions". Annales de l'Institut Fourier 62 (5): 1819–1887. 2012. doi:10.5802/aif.2737. http://www.numdam.org/item/AIF_2012__62_5_1819_0.
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