Mean-periodic function

Summary

A mean-periodic function is a continuous function on the real line that satisfies a convolution equation $f * \mu = 0$ for some nonzero compactly supported measure $\mu$. A periodic function with period $p$ is the case $\mu = \delta_0 - \delta_p$, and exponentials $x^k e^{\lambda x}$ are also mean-periodic. Equivalently, the translates of $f$ do not span a dense subspace of the continuous functions. Laurent Schwartz proved that every mean-periodic function on $\R$ is a limit of exponential polynomials taken from the closed span of its translates.