Abstract
We study the rate of convergence of Fourier series of orthogonal polynomials over an area inside and on the closure of regions of the complex plane.
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REFERENCES
C. N. Mergelyan, “On the completeness of systems of analytic functions,” Usp. Mat. Nauk., 8, No. 4(56), 3–63 (1953).
P. K. Suetin, “Polynomials orthogonal over a region and Bieberbach polynomials,” Proc. Steklov Inst. Math., 100 (1974).
F. G. Abdullayev, “Uniform convergence of the generalized Bieberbach polynomials in the regions with non-zero angles,” Acta Math. (Hung.), 77, 223–246 (1997).
F. G. Abdullaev, “On the speed convergence of Fourier series of orthogonal polynomials in domains with piecewise-quasiconformal boundary,” in: Theory of Mappings and Approximation of Functions, Naukova Dumka, Kiev (1989), pp. 3–11.
F. G. Abdullaev and V. V. Andrievskii, “On orthogonal polynomials over domains with K-quasiconformal boundary,” Izv. Akad. Nauk. Azerb. SSR, Ser. F.T.M., No. 1, 3–7 (1983).
F. G. Abdullaev, “On the convergence of Fourier series of orthogonal polynomials in domains with arbitrary K-quasiconformal boundary,” Izv. Akad. Nauk. Azerb. SSR, Ser. F.T.M., No. 4, 3–7 (1983).
C. Pommerenke, Univalent Functions, Vandenhoeck & Ruprecht, Göttingen, 1975.
D. Gaier, “Estimates of conformal mappings near the boundary,” Indiana Univ. Math. J., 21, 581–595 (1972).
F. D. Lesley, “Hölder continuity of conformal mappings at the boundary via the strip method,” Indiana Univ. Math. J., 31, 341–354 (1982).
S. E. Warschawskii, “On differentiability at the boundary in conformal mapping,” Proc. Amer. Math. Soc., 12, 614–620 (1961).
S. E. Warschawskii, “On the Hölder continuity at the boundary in conformal maps,” J. Math. Mech., 18, 423–427 (1968).
V. V. Andrievskii, B. I. Belyi, and V. K. Dzyadyk, Conformal Invariants in Constructive Theory of Functions of Complex Variable, World Federation, Atlanta (1995).
F. G. Abdullaev, “On the interference of the weight and boundary contour for orthogonal polynomials over the region,” J.C.A.A.A. (to appear).
V. V. Andrievskii, “Constructive characterization of harmonic functions in domains with quasiconformal boundary,” in: Quasiconformal Continuation and Approximation by Functions in a Set of the Complex Plane [in Russian], Kiev (1985), pp. 3–14.
L. V. Ahlfors, Lectures on Quasiconformal Mappings, van Nostrand, New YorkJędrysekPrinceton, NJ (1966).
D. Gaier, “On the convergence of the Bieberbach polynomials in regions with corners,” Const. Approxim., 4, 289–305 (1988).
V. I. Belyi, “Conformal mappings and approximation of analytic functions in domains with quasiconformal boundary,” Mat. Sb., 289–317 (1977).
J. M. Anderson, F. W. Gehring, and A. Hinkkanen, “Polynomial approximation in quasidisks,” Different. Geom. Complex Analysis, 75–86 (1985).
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Abdullaev, F.G., Küçükaslan, M. On the Convergence of Fourier Series with Orthogonal Polynomials inside and on the Closure of a Region. Ukrainian Mathematical Journal 54, 1567–1582 (2002). https://doi.org/10.1023/A:1023705500910
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DOI: https://doi.org/10.1023/A:1023705500910