Best Meromorphic Approximation of Markov Functions on the Unit Circle

Abstract.

Let E \subset(-1,1) be a compact set, let μ be a positive Borel measure with support \supp μ =E , and let H_p(G),1≤ p ≤∈fty, be the Hardy space of analytic functions on the open unit disk G with circumference Γ={z \colon |z|=1} . Let Δ_n,p be the error in best approximation of the Markov function \frac{1}{2π i} ∈t_E \frac{d μ(x)}{z-x} in the space L _p(Γ) by meromorphic functions that can be represented in the form h=P/Q, where P ∈ H_p(G),Q is a polynomial of degree at most n, Q\not \equiv 0. We investigate the rate of decrease of Δ_n,p,1≤ p ≤∈fty, and its connection with n -widths. The convergence of the best meromorphic approximants and the limiting distribution of poles of the best approximants are described in the case when 1<p≤∈fty and the measure μ with support E=[a,b] satisfies the Szegő condition ∈t_a^b \frac{\log(d μ/ d x)}{\sqrt{(x-a)(b-x)}} dx >- ∈fty.