On the Distribution of Zeros and Poles of Rational Approximants on Intervals

V V Andrievskii,R K Kovacheva,H.-P Blatt

doi:10.1155/2012/961209

Abstract

The distribution of zeros and poles of best rational approximants is well understood for the functions , . If is not holomorphic on , the distribution of the zeros of best rational approximants is governed by the equilibrium measure of under the additional assumption that the rational approximants are restricted to a bounded degree of the denominator. This phenomenon was discovered first for polynomial approximation. In this paper, we investigate the asymptotic distribution of zeros, respectively, -values, and poles of best real rational approximants of degree at most to a function that is real-valued, but not holomorphic on . Generalizations to the lower half of the Walsh table are indicated.

Highlights

Let B be a subset of C; we denote by m1 B : inf |Uν|the m1-measure of B, where the infimum is taken over all coverings {Uν} of B by disks Uν and |Uν| is the radius of the disk Uν.Let D be a region in C and φ a function defined in D with values in C
The normalized zero counting measures of rn∗,0 f converge in the weak∗-sense to the equilibrium measure of −1, 1, at least for a subsequence n ∈ Λ ⊂ N 4. This result was generalized to rational approximation with a bounded number of poles cf. 5, Theorem 4.1
The aim of the present paper is to investigate the distribution of the zeros of the rational approximants via the distribution of the alternation points

Summary

Introduction

1.1 ν the m1-measure of B, where the infimum is taken over all coverings {Uν} of B by disks Uν and |Uν| is the radius of the disk Uν. Let {rn}n∈N, rn ∈ Rn,n, be a sequence of rational functions converging geometrically to a function f on S, that is, lim sup f − rn. 1.6 for each compact set K ⊂ D, the sequence {rn}n∈N converges m1-almost geometrically inside D to a meromorphic function f ∈ Mm D. The normalized zero counting measures of rn∗,0 f converge in the weak∗-sense to the equilibrium measure of −1, 1 , at least for a subsequence n ∈ Λ ⊂ N 4 This result was generalized to rational approximation with a bounded number of poles cf 5, Theorem 4.1. The aim of the present paper is to investigate the distribution of the zeros of the rational approximants via the distribution of the alternation points

Main Results

Auxiliary Tools

Proofs

Generalization to the Lower-Half of the Walsh Table

Remarks