Asymptotic Error Estimates for L2 Best Rational Approximants to Markov Functions

Let f (z)= (t&z) d+(t) be a Markov function, where + is a positive measure with compact support in R. We assume that supp(+)/(&1, 1), and investigate the best rational approximants to f in the Hardy space H 2(V), where V :=[z # C | |z|>1] and H 2(V) is the subset of functions f # H2(V) with f ( )=0. The central topic of the paper is to obtain asymptotic error estimates for these approximants. The results are presented in three groups. In the first one no specific assumptions are made with respect to the defining measure + of the function f. In the second group it is assumed that the measure + is not too thin anywhere on its support so that the polynomials pn , orthonormal with respect to the measure +, have a regular n th root asymptotic behavior. In the third group the defining measure + is assumed to belong to the Szego class. For each of the three groups, asymptotic error estimates are proved in the L-norm on the unit circle and in a pointwise fashion. Also the asymptotic distribution of poles, zeros, and interpolation points of the best L approximants are studied. 2001 Academic Press