# Mean-periodic function

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In mathematical analysis, the concept of a **mean-periodic function** is a generalization introduced by Jean Delsarte, of the concept of a periodic function.[1]

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

where is the difference between the Dirac measures at 0 and *a*. A mean-periodic function is a function *ƒ* satisfying (1) for some nonzero measure with compact (hence bounded) support.