Mean-periodic function

Further results were made by Laurent Schwartz and J-P Kahane.

is some arbitrary nonzero measure with compact (hence bounded) support.

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).

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.

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