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  "body": "\n## Summary\n\nA Besicovitch almost periodic function is a locally integrable function on the real line with the following property.\nIt can be approximated in a mean sense by trigonometric polynomials with real frequencies.\nThe mean is the upper average of $\\abs{f}^p$ over long intervals $[-T, T]$.\nThis class, introduced by Abram Besicovitch, is much larger than the class of [[Bohr almost periodic functions|ED69H4C3PK]].\nFor $p = 2$ it carries a complete Fourier theory with a Parseval identity, and it is used for signals with discrete but irregular spectrum.\n\n## Explanations\n\n### L1 human\n\nBohr's almost periodic functions must be uniformly close to sums of waves everywhere on the line.\nBesicovitch only asks that they be close on average over long stretches of the line.\nSo a function may have jumps, or even large values on a bounded region, since a bounded region contributes nothing to a long-run average.\nFor instance, the indicator function of $[0, 1]$ is Besicovitch almost periodic, because its long-run average size is $0$.\n",
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