---
tag: 3F9G3DRQ62
kind: concept
name: Orthogonal polynomials on the unit circle
msc: [42C05, 33C47]
sources:
  - url: https://en.wikipedia.org/wiki/Orthogonal_polynomials_on_the_unit_circle
same_as:
  wikidata: Q7104532
  wikipedia: Orthogonal polynomials on the unit circle
---

## Summary

Orthogonal 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.
They are the circle analogue of [[orthogonal polynomials|8TXASARQN1]] on the real line.
Instead of a three-term recurrence they satisfy the Szegő recurrence, governed by a sequence of numbers in the unit disk, the Verblunsky coefficients.
The theory, started by Gábor Szegő, is linked to [[Toeplitz matrices|GKY2WSHE26]], to prediction theory of stationary processes, and to spectral theory of CMV matrices.

## Explanations

### L1 human

On the unit circle, the powers $z^n = e^{in\theta}$ are orthogonal for the uniform measure, which is the basis of Fourier series.
If the circle carries a different weight, the powers are no longer orthogonal.
Straightening them one at a time with respect to the new weight gives a new sequence of polynomials, one of each degree.
These polynomials record the weight in a form well suited to computation and to asymptotic questions.