Skip to main content
Log in

On the Convergence of Fourier Series with Orthogonal Polynomials inside and on the Closure of a Region

  • Published:
Ukrainian Mathematical Journal Aims and scope

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

REFERENCES

  1. C. N. Mergelyan, “On the completeness of systems of analytic functions,” Usp. Mat. Nauk., 8, No. 4(56), 3–63 (1953).

    Google Scholar 

  2. P. K. Suetin, “Polynomials orthogonal over a region and Bieberbach polynomials,” Proc. Steklov Inst. Math., 100 (1974).

  3. F. G. Abdullayev, “Uniform convergence of the generalized Bieberbach polynomials in the regions with non-zero angles,” Acta Math. (Hung.), 77, 223–246 (1997).

    Google Scholar 

  4. 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.

    Google Scholar 

  5. 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).

  6. 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).

  7. C. Pommerenke, Univalent Functions, Vandenhoeck & Ruprecht, Göttingen, 1975.

    Google Scholar 

  8. D. Gaier, “Estimates of conformal mappings near the boundary,” Indiana Univ. Math. J., 21, 581–595 (1972).

    Google Scholar 

  9. F. D. Lesley, “Hölder continuity of conformal mappings at the boundary via the strip method,” Indiana Univ. Math. J., 31, 341–354 (1982).

    Google Scholar 

  10. S. E. Warschawskii, “On differentiability at the boundary in conformal mapping,” Proc. Amer. Math. Soc., 12, 614–620 (1961).

    Google Scholar 

  11. S. E. Warschawskii, “On the Hölder continuity at the boundary in conformal maps,” J. Math. Mech., 18, 423–427 (1968).

    Google Scholar 

  12. V. V. Andrievskii, B. I. Belyi, and V. K. Dzyadyk, Conformal Invariants in Constructive Theory of Functions of Complex Variable, World Federation, Atlanta (1995).

    Google Scholar 

  13. 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).

  14. 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.

  15. L. V. Ahlfors, Lectures on Quasiconformal Mappings, van Nostrand, New YorkJędrysekPrinceton, NJ (1966).

    Google Scholar 

  16. D. Gaier, “On the convergence of the Bieberbach polynomials in regions with corners,” Const. Approxim., 4, 289–305 (1988).

    Google Scholar 

  17. V. I. Belyi, “Conformal mappings and approximation of analytic functions in domains with quasiconformal boundary,” Mat. Sb., 289–317 (1977).

  18. J. M. Anderson, F. W. Gehring, and A. Hinkkanen, “Polynomial approximation in quasidisks,” Different. Geom. Complex Analysis, 75–86 (1985).

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

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

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1023705500910

Keywords

Navigation