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  "body": "\n## Summary\n\nAn almost periodic function is a function that nearly repeats itself.\nFor any tolerance there are many shifts that change it by less than the tolerance.\nOn the real line, the uniformly almost periodic functions are exactly the uniform limits of finite sums $\\sum c_k e^{i\\lambda_k x}$ with arbitrary real frequencies $\\lambda_k$.\nThey extend the theory of Fourier series from [[periodic functions|0ATF6Q0633]] to functions such as $\\cos x + \\cos(\\sqrt{2} x)$.\nThe notion was created by Harald Bohr, and it extends to abelian groups, where it is linked to the [[Bohr compactification|G852FHZ3W1]].\n\n## Explanations\n\n### L1 human\n\nThe function $\\cos x + \\cos(\\sqrt{2} x)$ is not periodic, because the two periods $2\\pi$ and $\\sqrt{2} \\cdot 2\\pi$ have an irrational ratio.\nStill, one can find shifts $\\tau$ for which both $\\tau / 2\\pi$ and $\\sqrt{2}\\tau / 2\\pi$ are very close to whole numbers.\nShifting by such a $\\tau$ changes the function by very little everywhere.\nFunctions with plenty of such near-periods are called almost periodic.\nThere are several precise versions: uniform ones due to Bohr, and mean-square ones due to Stepanov, Weyl and Besicovitch.\n",
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