Why an orthonormal polynomial set over a continuous domain is not over a discrete one?

Question

I would like to read the proof showing that a orthonormal polynomial set over a continuous domain is neither orthonormal nor complete over a discrete values on that domain.

For example, Zernike polynomials are a complete and orthonormal set of polynomials over the continuous unit disk. However, they are not over a set of discrete samples in the unit disk. Where can I read why?

Thanks for your time.

Answer 1

There is a theory concerning discrete measures. See for example the book "Discrete orthogonal polynomials : asymptotics and applications" by J. Baik, T. Kriecherbauer, K. T.-R. McLaughlin & P. D. Miller. P. 8-9 may be what you are looking for.

Also, there is a theory of orthogonal polynomials for measures with a support equal to a Cantor set, hence a totally disconnected set. See for example, "Orthogonal polynomials associated with invariant measures on Julia sets" and "Infinite-dimensional jacobi matrices associated with Julia sets" by M. F. Barnsley, J. S. Gerónimo and A. N. Harrington.