Computation of orthogonal polynomials and rational approximants in the chebfun system

Chebfun is a new class in object-oriented Matlab that approximates functions very accurately by using methods based on Chebyshev expansions and interpolation. Chebfuns can be used to explore and obtain new insights on computational aspects of approximation theory and in this talk we present chebfun-based methods for the construction of orthogonal polynomials and rational approximations. In particular, we explain how to construct sets of orthogonal polynomials by means of Lanczos iterations on quasi-matrices. We also present best rational approximations and near-best rational approximations obtained with the implementation of a chebfun-Remez algorithm and a chebfun-Carathéodory-Fejér algorithm respectively. We finish this talk by discussing the construction of Padé and Chebyshev-Padé approximates using chebfuns, and some specific properties of these approximations.