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.