Mean-periodic function

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

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

where $\backslash mu$ is the difference between the Dirac measures at 0 and a. The function {{math|f}} is mean-periodic if it satisfies the same equation (1), but where $\backslash mu$ 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 {{math|f}} for which there exists a compactly supported (signed) Borel measure $\backslash mu$ for which $f*\backslash mu\; =\; 0$.

There are several well-known equivalent definitions.

Mean-periodic functions are a separate generalization of periodic functions from the almost periodic functions. For instance, exponential functions are mean-periodic since {{math|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.

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.{{Cite journal |last=Laird |first=P. G. |date=1972 |title=Some properties of mean periodic functions |url=http://dx.doi.org/10.1017/s1446788700011058 |journal=Journal of the Australian Mathematical Society |volume=14 |issue=4 |pages=424–432 |doi=10.1017/s1446788700011058 |issn=0004-9735}}

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.{{cite journal|author3-first = M.|author3-last = Suzuki|author2-first=G.|author2-last=Ricotta|author1-first=I.|author1-last=Fesenko|author1-link=Ivan Fesenko|title=Mean-periodicity and zeta functions|journal=Annales de l'Institut Fourier|year=2012|volume=62|pages=1819–1887|number=5|url=http://www.numdam.org/item/AIF_2012__62_5_1819_0|arxiv=0803.2821|doi = 10.5802/aif.2737}} There is a certain class of mean-periodic functions arising from number theory.

• 　almost periodic functions

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Category:Mathematical analysis