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  "body": "\n## Summary\n\nA 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 $\\mu$.\nA [[periodic function|0ATF6Q0633]] with period $p$ is the case $\\mu = \\delta_0 - \\delta_p$, and exponentials $x^k e^{\\lambda x}$ are also mean-periodic.\nEquivalently, the translates of $f$ do not span a dense subspace of the continuous functions.\nLaurent Schwartz proved that every mean-periodic function on $\\R$ is a limit of exponential polynomials taken from the closed span of its translates.\n\n## Explanations\n\n### L1 human\n\nA periodic function equals its own shift: $f(x) - f(x - p) = 0$.\nA mean-periodic function satisfies a weaker rule: some fixed weighted average of its shifted copies is always zero.\nFor example, $e^{\\lambda x}$ satisfies $f(x) - e^{\\lambda} f(x - 1) = 0$, although it is not periodic for real $\\lambda \\ne 0$.\nSuch functions behave like sums of exponential waves, which explains the name.\n",
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