Asymptotic Error Estimates for L2 Best Rational Approximants to Markov Functions

Abstract

Let f (z)= (t&z)&1 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 0 2 (V), where V :=[z # C | |z|>1] and H 0 2 (V) is the subset of functions f # H 2 (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 p n , 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 2 -norm on the unit circle and in a pointwise fashion. Also the asymptotic distribution of poles, zeros, and interpolation points of the best L 2 approximants are studied. Academic PressKey Words: best rational approximation in the L 2 -norm on the unit circle; asymptotic error estimates; Markov's theorem.

The Riemann–Hilbert approach to strong asymptotics for orthogonal polynomials on [−1,1]

“…We found it more convenient to use (1.6). This form was also used, for example, in [6,7]. Remark 1.2 There are also results for the orthogonal polynomials on the interval [−1 , 1] under conditions that are somewhat stronger than the Szegő condition, see e.g.…”

On rational approximation of Markov functions on finite sets

“…We also mention papers of Anderson [4], Braess [13], Baratchart, Prokhorov and Saff [6][7][8], Barrett [10], Pekarskii [23] related to rational and meromorphic approximation of Markov functions. In [9] Baratchart, Stahl and Wielonsky gave sharp error rates for the best H 2 approximants (particular Padé approximants) to Markov functions. In the book of Braess [12] some properties of best rational approximants and multipoint Padé approximants for Markov functions are proved, including existence and uniqueness of the corresponding approximants and the absence of defect.…”

Zero distributions via orthogonality

“…Although (2) looks like ordinary orthogonality with varying weight Q −2 n (t), it is not so because this weight itself depends on q n . When λ is a positive measure on [a, b] ⊂ (−1, 1) satisfying the Szegő condition, then the asymptotic zero distribution of the polynomials q n in (2) is known (and much beyond, see [2], [5]), but still our result deals with less regular situations. And when λ is complex, the theorem we prove is first of this kind.…”