On the existence of compacta of minimal capacity in the theory of rational approximation of multi-valued analytic functions

Abstract

АннотацияFor an interval E = [a, b] on the real line, let µ be either the equilibrium measure, or the normalized Lebesgue measure of E, and let V µ denote the associated logarithmic potential. In the present paper, we construct a function f which is analytic on E and possesses four branch points of second order outside of E such that the family of the admissible compacta of f has no minimizing elements with regard to the extremal theoretic-potential problem, in the external field equals V −µ .Bibliography: 35 items.rational approximants, Padé approximation, orthogonal polynomials, distribution of poles, convergence in capacity Dedicated to the memory of Andrei Aleksandrovich Gonchar and Herbert Stahl NotationsThroughout the paper, we use the following notations.

Capacity of a compact set in a logarithmic potential field

“…Another theorem proved in this paper extends the result from [13] on the existence of functions with four second-order branch points for which the existence problem has no positive solution to the case of functions that have only two second-order branch points. Namely, we prove the following theorem.…”

“…Moreover, it turns out that even for the simplest (not degenerate to a point) continuum E = [−1, 1] and the equilibrium measure μ (or the normalized Lebesgue measure and other measures of sufficiently general type) on E, there exist functions f ∈ A(E) such that equality (1.7) holds for no compact set F ∈ K E,f . In [13], one of such functions is indicated; it has four second-order branch points, of which two lie in {|z| ≤ r} ∩ {Im z > 0} and the other two lie in {|z| ≤ r} ∩ {Im z < 0}, where r is a sufficiently small positive number.…”

“…In more general cases, one usually applies the potential-theoretic variational method developed by Rakhmanov [10] (see also [11,12]). In the form presented above, the theorem was formulated and proved in [13].…”

О сходимости $m$-точечных аппроксимаций Паде набора многозначных аналитических функций

“…Наша цель -доказать, что эта точная нижняя грань дости-гается на некотором компакте F ∈ K f . (Как показывает построенный в [16] пример функции f ∈ H([−1, 1]), аналитической в C \ {a 1 , a 2 , a 3 , a 4 }, нижняя точная грань может не достигаться в случае, когда µ -равновесная мера или нормированная мера Лебега на [−1, 1]).…”

“…Полное доказательство этих импликаций, при-чем в более общей ситуации, см. в [16].) Как отмечалось выше, утверждение 5 теоремы следует из утверждений 3 и 4 в силу доказанной в [9] общей теоремы, сформулированной в конце первого параграфа.…”