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Various important sets of polynomials (Legendre, Chebyshev, etc.) are orthogonal over some real interval with some weighting. Are there known families of polynomials that are orthogonal over other curves in the complex plane?

For instance, I would like a basis for the polynomials of degree n that is orthogonal over, say, the circle

$$-1 + \exp(it)$$

for $0\le t< 2\pi$.

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Here are some nice links and references to the theory of orthogonal polynomials over the unit circle. –  Ragib Zaman Nov 30 '11 at 5:59
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Gabor Szegő did study a family of orthogonal polynomials that are orthogonal with respect to the unit circle; see this for instance. –  J. M. Nov 30 '11 at 6:03

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up vote 4 down vote accepted

Gabor Szegő, in his nice book on orthogonal polynomials, discusses in one chapter a procedure for generating polynomials orthogonal with respect to the inner product

$$\langle f,g\rangle=\frac1{L}\oint_\gamma w(t) f(t)\overline{g(t)}|\mathrm dt|$$

where $\gamma$ is an arc in the complex plane, $L$ is the length of the curve, $w(t)$ is a positive and continuous weight function defined on $\gamma$, and $|\mathrm dt|$ is the arc element on $\gamma$. One can do an orthogonalization method like Gram-Schmidt using this inner product. See Szegő's book and Saff's survey, among a number of references. Parts of this article discuss the extension of Gaussian quadrature to inner products over arcs.

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