Ralitza K. Kovacheva

Poles and Alternation Points in Real Rational Chebyshev Approximation

The distribution of equi-oscillation points (alternation points) for the error in best Chebyshev approximation on [−1,1] by rational functions is investigated. In general, the alternation points need not be dense in [−1,1] when rational functions of degree (n, m) are considered and asymptotically n/m → κ with κ ≥ 1. We show that the asymptotic behavior of the alternation points is closely related to the behavior of the poles of the rational approximants. Hence, poles of the rational approximations are attracting points of alternations such that the well-known equi-distribution for the polynomial case can be heavily disturbed.

Distribution of Zeros of the Hermite Pade Polynomials for a System of Three Functions, and the Nuttall Condenser

The well-known approach of J. Nuttall to the derivation of strong asymptotic formulas for the Hermite-Padé polynomials for a set of m multivalued functions is based on the conjecture that there exists a canonical (in the sense of decomposition into sheets) m-sheeted Riemann surface possessing certain properties. In this paper, for m = 3, we introduce a notion of an abstract Nuttall condenser and describe a procedure for constructing (based on this condenser) a three-sheeted Riemann surface

Distribution of interpolation points of maximally convergent multipoint Padé approximants

Re _3

Distribution of points of interpolation of multipoint Padé approximants

that has a canonical decomposition. We consider a system of three functions

An Analogue of Montel's Theorem for Rational Functions of Best Lp- Approximation

mathfrak{f}_1 ,mathfrak{f}_2 ,mathfrak{f}_3

APPROXIMATION ON AN UNBOUNDED INTERVAL

that are rational on the constructed Riemann surface and satisfy the independence condition det . In the case of m = 3, we refine the main theorem from Nuttall’s paper of 1981. In particular, we show that in this case the complement ℂ̄ B of the open (possibly, disconnected) set B ⊂ ℂ̄ introduced in Nuttall’s paper consists of a finite number of analytic arcs. We also propose a new conjecture concerning strong asymptotic formulas for the Padé approximants.