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  "body": "\n## Summary\n\nThe prolate spheroidal wave functions are the bounded solutions on $[-1, 1]$ of a second-order differential equation that arises from separating variables in prolate spheroidal coordinates.\nD. Slepian and H. O. Pollak found that the same functions are eigenfunctions of a time and band limiting operator.\nThis operator first restricts a function to $[-1, 1]$ and then removes all frequencies outside a fixed band.\nSo they are the [[band-limited functions|P3T104G564]] that are most concentrated on an interval.\nThey answer the question of how much of its energy a band-limited signal can keep in a given time window.\nThey are used in signal processing and spectral estimation.\nSee the [[definition|SDYEK7HEZQ]].\n\n## Explanations\n\n### L1 human\n\nNo nonzero signal can be both limited in time and limited in frequency.\nOne can still ask which band-limited signal puts as large a share of its energy as possible inside a given time interval.\nThe answer is the first prolate spheroidal wave function; the next ones give the best signals that are orthogonal to the previous ones.\nWhen the product of the time interval and the frequency band is large, many of them keep almost all their energy inside the interval.\nTheir number is roughly proportional to that product.\n",
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