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  "body": "\n## Summary\n\nOrthogonal polynomials on the unit circle are the polynomials obtained by Gram–Schmidt orthogonalization of $1, z, z^2, \\dots$ in $L^2$ of a probability measure on the unit circle.\nThey are the circle analogue of [[orthogonal polynomials|8TXASARQN1]] on the real line.\nInstead of a three-term recurrence they satisfy the Szegő recurrence, governed by a sequence of numbers in the unit disk, the Verblunsky coefficients.\nThe theory, started by Gábor Szegő, is linked to [[Toeplitz matrices|GKY2WSHE26]], to prediction theory of stationary processes, and to spectral theory of CMV matrices.\n\n## Explanations\n\n### L1 human\n\nOn the unit circle, the powers $z^n = e^{in\\theta}$ are orthogonal for the uniform measure, which is the basis of Fourier series.\nIf the circle carries a different weight, the powers are no longer orthogonal.\nStraightening them one at a time with respect to the new weight gives a new sequence of polynomials, one of each degree.\nThese polynomials record the weight in a form well suited to computation and to asymptotic questions.\n",
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