Approximations of algebraic functions by rational ones – functional analogues of diophantine approximants

Asymptotics of Polynomials Orthogonal on a Cross with a Jacobi-Type Weight

“…k ∈ {0, 1}, where g ∆ (z; ∞) is the Green function for C \ ∆ with pole at ∞. 2 In fact, the above properties allow us to verify that…”

“…This study is not only motivated by the authors' intrinsic interest in the behavior of non-Hermitian orthogonal polynomials, but also by the research program of determining the effective constants in the functional analog of Thue-Siegel-Roth theorem, see [2] for a more detailed explanation. Briefly, let f (z) = ∞ i=0 f i z −i be a convergent power series.…”

Continued Fractions and Conformal Mappings for Domains with Angle Points

“…The continued fraction converges to √ x at x = 0 faster than the Taylor expansion of the function x−a a + 1 into degrees of (x − a) to the function x−a a + 1 itself at x = 0 [5]. The most thorough and refined method here is the Padé rational function approximation of the algebraic function [3,4]. Note that these approximations are optimal in the set of fraction polynomials though their construction requires application of Euclidean algorythm and additional investigation of the holomorphness domain D.…”

Continued Fractions and Conformal Mappings for Domains with Angel Points

“…Consider a curve whose behavior at an angle point is similar to Thus, the method from [19] is difficult to apply, since even the Fourier series poorly approximate a curve with such an angle point. Let the domain boundary be angled and the angle be equal to The most thorough and refined method here is the Padé rational function approximation of the algebraic function [3,4]. Note that these approximations are optimal in the set of fraction polynomials though their construction requires application of Euclidean algorythm and additional investigation of the holomorphness domain D .…”