Approximation and Capacities

The Gonchar-Stahl $ \rho^2$-theorem and associated directions in the theory of rational approximations of analytic functions

“…for any continuum E in the domain of the function (see [17], [18], [25], [28]). In other words, if lim ρ n (f, E) 1/n exists then it is equal to ρ(f ) 2 .…”

Padé approximants, continued fractions, and orthogonal polynomials

“…The foundations of the modern theory of Padé approximants were laid in the 1970s-80s, in the papers of Gonchar [19]- [31] and Nuttall [32]- [40]. The results and open problems formulated by them in that period determined the main directions of research in the area of constructive rational approximation for many years.…”

“…Several generalizations of the classical Padé approximants have been considered: multipoint Padé approximants, Padé approximants to orthogonal expansions, to Faber series and trigonometric series, Hermite-Padé approximation, and others (see [8], [11], [12], [26], [30], [31], [97], [123], [124], [132]- [139]). All these cases are concerned with approximation by rational functions with free poles, and the approximants themselves have a non-linear character.…”

“…be the distance of the function f n in the uniform norm on the compact set E from the class R n of rational functions r of degree n. The present chapter is devoted to the rate of rational approximation of the functions f n , that is, the asymptotic behaviour of ρ n as n → ∞. The modern theory of this classical problem was developed by Gonchar (see [19]- [31], [47]- [49], [71], [132]- [135], [137], [182], which is a far from complete list of key contributions). Under the most general assumptions about F , E, and Φ n the result describing the order of the rate of rational approximation (122) was obtained in the paper [49] of Gonchar and Rakhmanov. In this chapter we also discuss a refinement of the result on the rate of approximation (122) for real-symmetric functions (120), approximated on intervals (finite or infinite) of the real line.…”

Аппроксимации Паде, Непрерывные Дроби И Ортогональные Многочлены

“…Основы современной теории аппроксимаций Паде были заложены в 70-80-х годах прошлого века в работах А. А. Гончара [19]- [31] и Дж. Наттолла [32]- [40].…”