Simultaneous approximation of the Riemann conformal map and its derivatives by Bieberbach polynomials

Yıl 2017, Cilt: 46 Sayı: 2, 209 - 216, 01.04.2017

Daniyal M. Israfilov

Öz

Let $G$ be a domain in the complex plane $\mathbb{C}$ bounded by a rectifiable Jordan curve $\Gamma$, let $z_0\in G$ and let $\varphi_0$ be the Riemann conformal map of $G$ onto $\mathbb{D}_r:=\{ w\in\mathbb{C}\,:\, |w|<r \}$, normalized by $\varphi_0(z_0)=0$, $\varphi_0'(z_0)=1$. In this work the simultaneous approximations of $\varphi_0$ and its derivatives by Bieberbach polynomials are investigated. The approximation rate in dependence of the smoothness parameters of the considered domains is estimated. 

Anahtar Kelimeler

Bieberbach polynomials, conformal map, simultaneous approximation

Kaynakça

• Abdullayev, F. G.: Uniform convergence of the Bieberbach polynomials inside and on the closure of domains in the complex plane. East J. Approx. 7(1), 77-101(2001)
• Abdullayev, F. G., Aral, N.: The Relation Between Different Norms of Algebraic Polynomials in the Regions of Complex Plane. Azerbaijan Journal of Mathematics, Vol. 1(2), 70-82(2011)
• Alper, S. Y.: Approximation in the mean of analytic functions of class Ep. (Russian), In: Investigations on the modern problems of the function theory of a complex variable, Moscow: Gos. Izdat. Fiz. Mat. Lit., 273-286(1960)
• Andrievskii, V. V.: Convergence of Bieberbach polynomials in domains with quasi-conformal boundary. Ukrainian Math. J. 35(3), 233-236(1983)
• Andrievskii, V. V., Gaier, D.: Uniform convergence of Bieberbach polynomials in domains with piecewise quasianalytic boundary. Mitt. Math. Sem. Giessen 211, 49-60(1992)
• Andrievskii, V. V., Pritsker, I. E.: Convergence of Bieberbach polynomials in domains with interior cusps. J. d'Analyse Math. 82, 315-332(2000)
• Andrievskii, V. V., Pritsker, I. E., Varga, R.: Simultaneous approximation and interpolation of functions on continua in the complex plane. J. Math. Pures Appl. 80, 373-388(2001)
• De Vore, R. A., Lorentz, G. G.: Constructive Approximation. Springer-Verlag, Berlin (1993)
• Dyn'kn, E. M.: The rate of polynomial approximation in the complex domain. In Complex analysis and spectral theory (Leningrad 1979-1980), Springer-Verlag, Berlin, 90-142(1981)
• Gaier, D.: Lectures on Complex Approximation. Birkhauser, Boston (1987)
• Gaier, D.: On the convergence of the Beberbach polynomials in regions with corners. Constr. Approximation, 4, 289-305(1988)
• Goluzin, G. M.: Geometric Theory of Functions of a Complex Variable.:Translation of Mathematical Monographs. Vol. 26: AMS; (1969)
• Israfilov, D. M.: Approximation by p-Faber polynomials in the weighted Lebesgue space $L^p(L,w)$ and the Bieberbach polynomials. Constr. Approx. 17, 335-351(2001)
• Israfilov, D. M.: Uniform convergence of the Bieberbach polynomials in closed smooth domains of bounded boundary rotation. Journal of Approx. Theory, 125, 116-130(2003)
• Keldych, M. V.: Sur I'appoximation en moyenne quadratique des fonctions analitiques. Math. Sb., 5 (47)), 391-401(1939)
• Mergelyan, S. N.: Certain questions of the constructive theory of functions. (Russian), Trudy Math, Inst. Steklov, 37, 1-91(1951)
• Pommerenke, Ch.: Boundary Behavior of Conformal Maps, Springer-Verlag, Berlin (1991)
• Simonenko, I. B.: On the convergence of Bieberbach polynomials in the case of a Lipschitz domain. Math. USSR-Inv., 13, 166-174(1978)
• Suetin, P. K.: Polynomials Orthogonal over a Region and Bieberbach Polynomials. Proceed- ings of the Steklov Institute of Mathematics, Amer. Math. Soc. Providence, Rhode Island, (1975)
• Warschawski, S.: Über das Randverhalten der Ableitung der Abbildungsfunktionen bei konformer Abbildung. Math. Z., 35, 321-456(1932)
• Wu Xue-Mou.: On Bieberbach polynomials. Acta Math. Sinica, 13, 145-151(1963)