# Polynomials that are orthogonal over curves in the complex plane

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 $0\le t< 2\pi$.

The reason that I'm posting this here is that I have a numerical problem involving a matrix of polynomial values over points in the complex plane. Using the monomial basis, it becomes very ill-conditioned for most sets of points. I'd like to use another basis to improve the conditioning, but it's not clear that using, say, Legendre or Chebyshev polynomials will improve the conditioning for general curves in the complex plane.

• I think your edit rendered almost my entire answer irrelevant :-P It's a better question now, though. – David Z Nov 30 '11 at 6:00
• I suspect that there is an appropriate modification of the Chebyshev algorithm for generating recursion coefficients. I gave a reference to Szegő in your math.SE question. – J. M. Nov 30 '11 at 6:59
• Thanks! Yes, this question was answered very well on math.SE, which is probably where I should have asked first. – David Ketcheson Nov 30 '11 at 15:14