On best rational approximation of analytic functions

Abstract

This paper contains some theorems related to the best approximation n (f ; E) to a function f in the uniform metric on a compact set E ⊂ C by rational functions of degree at most n. We obtain results characterizing the relationship between n (f ; K) and n (f ; E) in the case when complements of compact sets K and E are connected, K is a subset of the interior of E, and f is analytic in and continuous on E.

On estimates for the ratio of errors in best rational approximation of analytic functions

“…Therefore it follows from (4.3) that Since the left-hand side of (4.6) does not depend on ε and ε 1 , the proof of Theorem A, in the special case, follows from the properties of capacities that (see [8], [14]) lim ε→0, ε 1 →1…”

“…Also note that ∂E stands for the boundary of the set E and by C(F, K) we mean the capacity of the condenser (F, K) for a pair of disjoint compact subsets of C (see, for example, [8] and [16] for more details and the exact definition). In [14], the second author proves the above inequality in the case where the complements of E and K are both connected. Therefore, Theorem A can be considered as the generalization of the result in [14] with no additional assumptions on the compact sets E and K. One immediate consequence of Theorem A is the following estimate for the lower limit of (ρ n (f ; K)/ρ n (f ; E)) 1/n as n → ∞.…”

“…In [14], the second author proves the above inequality in the case where the complements of E and K are both connected. Therefore, Theorem A can be considered as the generalization of the result in [14] with no additional assumptions on the compact sets E and K. One immediate consequence of Theorem A is the following estimate for the lower limit of (ρ n (f ; K)/ρ n (f ; E)) 1/n as n → ∞.…”

Numerical solution of a class of complex differential equations by the Taylor collocation method in elliptic domains

“…In recent years, the studies on complex differential equations, i.e., a geometric approach based on meromorphic function in arbitrary domains [3], a topological description of solution of some complex differential equations with multivalued coefficients [4], the zero distrubition [5], and growth estimates [6] of linear complex differential equations, the rational and polynomial approximations of analytic functions in the complex plane [7,8], are developed very rapidly and intensively.…”

A collocation method to solve higher order linear complex differential equations in rectangular domains

“…In recent years, the studies on complex differential equations, i.e., a geometric approach based on meromorphic function in arbitrary domains [3], a topological description of solutions of some complex differential equations with multivalued coefficients [4], the zero distribution [5] and growth estimates [6] of linear complex differential equations, the rational and polynomial approximations of analytic functions in the complex plane [7,8], are developed very rapidly and intensively.…”