Uniform Convergence of Extremal Polynomials When Domains Have Corners and Special Cusps on the Boundary

ABSTRACT

We study the approximation properties of the extremal polynomials in Ap?norm and C?norm. We prove estimates for the rate of such convergence of the sequence of the extremal polynomials on domains with corners and special cusps.

We study the approximation properties of the extremal polynomials in Ap?norm and C?norm. We prove estimates for the rate of such convergence of the sequence of the extremal polynomials on domains with corners and special cusps.

Cite this paper

nullF. Abdullayev, C. Koşar and M. Kucukaslan, "Uniform Convergence of Extremal Polynomials When Domains Have Corners and Special Cusps on the Boundary,"*Advances in Pure Mathematics*, Vol. 1 No. 6, 2011, pp. 305-314. doi: 10.4236/apm.2011.16056.

nullF. Abdullayev, C. Koşar and M. Kucukaslan, "Uniform Convergence of Extremal Polynomials When Domains Have Corners and Special Cusps on the Boundary,"

References

[1] P. J. Davis, “Interpolation and Approximation,” Blaisdell Pub. Company, New York, 1963.

[2] M. Kucukaslan and F. G. Abdullayev, “New Extremal Polynomials and Its Approximations Properties,” Novisad Journal of Mathematics, Vol. 39, No. 2, 2009, pp. 1-12.

[3] M. V. Keldysh, “Sur L'approximation en Moyenne Quadratique des Fontions Analtiques,” Matematicheskii Sbornik, Vol. 5, No. 2, 1939, pp. 391-401.

[4] P. K. Suetin, “Polynomials Orthogonal over a Region and Bieberbach Polynomial,” American Mathematical Society, Rohde Island, 1974.

[5] S. N. Mergelyan, “Certain Questionsof the Constructive Theory of Functions,” Trudy Matematicheskogo Instituta im. V.A. Steklova RAN, Vol. 37, 1951, pp. 3-91.

[6] I. B. Simonenko, “On the Convergence of Bieberbach Polynomials in the Case of Lipschitz Domain,” Mathematics of the USSR-Izvestiya, Vol. 13, No. 1, 1979, pp. 166-174. doi:10.1070/IM1979v013n01ABEH002017

[7] V. V. Andrievskii, “On the Uniform Convergence of Bieberbach Polynomials in Domains with Piecewise Quasiconformal Boundary,” In: Mappings Theory and Approximation of Function, Naukova Dumka Kiev, 1983, pp. 3-18.

[8] V. V. Andrievskii, “Convergence of Bieberbach Polynomials in Domains with Quasiconformal Boundary,” Ukrainian Mathematical Journal, Vol. 35, No. 3, 1983, pp. 233-236. doi:10.1007/BF01092167

[9] D. Gaier, “Polynoimal Approximation of Conformal Maps,” Constructive Approximation, Vol. 14, 1994, pp. 27-40. doi:10.1007/s003659900061

[10] D. Gaier, “On the Convergence of the Bieberbach Polynomials in Regions with Corners,” Constructive Approximation, Vol. 4, 1998, pp. 289-305. doi:10.1007/BF02075463

[11] F. G. Abdullayev, “On the Convergence of Bieberbach Polynomials in Domains with Interior Zero Angles,” Dokladi Akademii Nauk Ukraine SSR Seria A, Vol. 12, 1989, pp. 3-5.

[12] F. G. Abdullayev, “Uniform Convergence of the Bieberbach Polynomials inside and on the closure of Domain in the Complex Plane,” East Journal on Approximations, Vol. 7, No. 1, 2001, pp. 77-101.

[13] F. G. Abdullayev and A. Baki, “On the Convergence of Bieberbach Polynomials in Domains with interior zero angles,” Complex Variables: Theory and Applications, Vol. 34, No. 2, 2001, pp. 131-143.

[14] D. Israfilov, “Approximation by p-Faber Polynomials in the weighted Simirnov class Ep(G,w) and the Bieberbach Polynomials,” Constructive Approximation, Vol. 17, No. 3, 2001, pp. 335-351. doi:10.1007/s003650010030

[15] D. Israfilov, “Uniform Convergence of the Bieberbach Polynomials in Closed Smooth Domains of Bounded Boundary Rotation,” Journal of Approximation Theory, Vol. 125, No. 1, 2003, pp. 116-130. doi:10.1016/j.jat.2003.09.008

[16] O. Lehto and K. I. Virtanen, “Quasiconformal Mapping in the Plane,” Springer Verlag, Berlin, 1973.

[17] S. Rickman, “Characterization of Quasiconformal Arcs,” Annales Academiae Scientiarum Fennicae Mathematica, 1996.

[18] F. G. Abdullayev, “On the Orthogonal Polynomials in Domains with Quasiconformal Boundary,” Dissertation, Donetsk (in Russian), 1986.

[19] L. V. Ahlfors, “Lectures on Quasiconformal Mappings,” Van Nostrand, Princeton, 1996.

[20] V. V. Andrievskii, V. I. Belyi, V. K. Dzjadyk, “Conformal Invariants in Constructive Theory of Functions of Complex Variable,” World Federation Pub., Atlanta, Georgia, 1995.

[21] F. G. Abdullayev, “Uniform Convergence of Generalized Bieberbach Polynomials in Regions with non-zero angles,” Acta Mathematica Hungarica, Vol. 77, No. 3, 1997, pp. 223-246.

[22] F. G. Abdullayev, “Uniform Convergence of Generalized Bieberbach Polynomials in Regions with Zero Angles,” Czechoslovak Mathematical Journal, Vol. 51, No. 3, 2001, pp. 643-660. doi:10.1023/A:1013796308878

[23] V. I. Belyi and I. E. Pritsker, “On the Curved Wedge Condition and the Continuity Moduli of Conformal Mapping,” Ukrainian Mathematical Journal, Vol. 45, No. 6, 1993, pp. 837-844.

[24] V. I. Simirnov and N. A. Lebedev, “Functions of Complex Variable Constructive Theory,” MIT Press, Cambridge, 1968.

[25] J. L. Walsh, “Interpolation and Approximation by Rational Functions in the Complex Domain,” American Mathematical Society, Rohde Island, 1960.

[26] F. G. Abdullayev and M. Kucukaslan, “On the Convergence of the Fourier Series of Orthonormal Polynmials in the Domain with Piecewise Smooth Boundary,” Proceeding of IMM of NAS of Azerbaijan, Vol. XIV, No. XXII, 2001, pp. 3-13.

[1] P. J. Davis, “Interpolation and Approximation,” Blaisdell Pub. Company, New York, 1963.

[2] M. Kucukaslan and F. G. Abdullayev, “New Extremal Polynomials and Its Approximations Properties,” Novisad Journal of Mathematics, Vol. 39, No. 2, 2009, pp. 1-12.

[3] M. V. Keldysh, “Sur L'approximation en Moyenne Quadratique des Fontions Analtiques,” Matematicheskii Sbornik, Vol. 5, No. 2, 1939, pp. 391-401.

[4] P. K. Suetin, “Polynomials Orthogonal over a Region and Bieberbach Polynomial,” American Mathematical Society, Rohde Island, 1974.

[5] S. N. Mergelyan, “Certain Questionsof the Constructive Theory of Functions,” Trudy Matematicheskogo Instituta im. V.A. Steklova RAN, Vol. 37, 1951, pp. 3-91.

[6] I. B. Simonenko, “On the Convergence of Bieberbach Polynomials in the Case of Lipschitz Domain,” Mathematics of the USSR-Izvestiya, Vol. 13, No. 1, 1979, pp. 166-174. doi:10.1070/IM1979v013n01ABEH002017

[7] V. V. Andrievskii, “On the Uniform Convergence of Bieberbach Polynomials in Domains with Piecewise Quasiconformal Boundary,” In: Mappings Theory and Approximation of Function, Naukova Dumka Kiev, 1983, pp. 3-18.

[8] V. V. Andrievskii, “Convergence of Bieberbach Polynomials in Domains with Quasiconformal Boundary,” Ukrainian Mathematical Journal, Vol. 35, No. 3, 1983, pp. 233-236. doi:10.1007/BF01092167

[9] D. Gaier, “Polynoimal Approximation of Conformal Maps,” Constructive Approximation, Vol. 14, 1994, pp. 27-40. doi:10.1007/s003659900061

[10] D. Gaier, “On the Convergence of the Bieberbach Polynomials in Regions with Corners,” Constructive Approximation, Vol. 4, 1998, pp. 289-305. doi:10.1007/BF02075463

[11] F. G. Abdullayev, “On the Convergence of Bieberbach Polynomials in Domains with Interior Zero Angles,” Dokladi Akademii Nauk Ukraine SSR Seria A, Vol. 12, 1989, pp. 3-5.

[12] F. G. Abdullayev, “Uniform Convergence of the Bieberbach Polynomials inside and on the closure of Domain in the Complex Plane,” East Journal on Approximations, Vol. 7, No. 1, 2001, pp. 77-101.

[13] F. G. Abdullayev and A. Baki, “On the Convergence of Bieberbach Polynomials in Domains with interior zero angles,” Complex Variables: Theory and Applications, Vol. 34, No. 2, 2001, pp. 131-143.

[14] D. Israfilov, “Approximation by p-Faber Polynomials in the weighted Simirnov class Ep(G,w) and the Bieberbach Polynomials,” Constructive Approximation, Vol. 17, No. 3, 2001, pp. 335-351. doi:10.1007/s003650010030

[15] D. Israfilov, “Uniform Convergence of the Bieberbach Polynomials in Closed Smooth Domains of Bounded Boundary Rotation,” Journal of Approximation Theory, Vol. 125, No. 1, 2003, pp. 116-130. doi:10.1016/j.jat.2003.09.008

[16] O. Lehto and K. I. Virtanen, “Quasiconformal Mapping in the Plane,” Springer Verlag, Berlin, 1973.

[17] S. Rickman, “Characterization of Quasiconformal Arcs,” Annales Academiae Scientiarum Fennicae Mathematica, 1996.

[18] F. G. Abdullayev, “On the Orthogonal Polynomials in Domains with Quasiconformal Boundary,” Dissertation, Donetsk (in Russian), 1986.

[19] L. V. Ahlfors, “Lectures on Quasiconformal Mappings,” Van Nostrand, Princeton, 1996.

[20] V. V. Andrievskii, V. I. Belyi, V. K. Dzjadyk, “Conformal Invariants in Constructive Theory of Functions of Complex Variable,” World Federation Pub., Atlanta, Georgia, 1995.

[21] F. G. Abdullayev, “Uniform Convergence of Generalized Bieberbach Polynomials in Regions with non-zero angles,” Acta Mathematica Hungarica, Vol. 77, No. 3, 1997, pp. 223-246.

[22] F. G. Abdullayev, “Uniform Convergence of Generalized Bieberbach Polynomials in Regions with Zero Angles,” Czechoslovak Mathematical Journal, Vol. 51, No. 3, 2001, pp. 643-660. doi:10.1023/A:1013796308878

[23] V. I. Belyi and I. E. Pritsker, “On the Curved Wedge Condition and the Continuity Moduli of Conformal Mapping,” Ukrainian Mathematical Journal, Vol. 45, No. 6, 1993, pp. 837-844.

[24] V. I. Simirnov and N. A. Lebedev, “Functions of Complex Variable Constructive Theory,” MIT Press, Cambridge, 1968.

[25] J. L. Walsh, “Interpolation and Approximation by Rational Functions in the Complex Domain,” American Mathematical Society, Rohde Island, 1960.

[26] F. G. Abdullayev and M. Kucukaslan, “On the Convergence of the Fourier Series of Orthonormal Polynmials in the Domain with Piecewise Smooth Boundary,” Proceeding of IMM of NAS of Azerbaijan, Vol. XIV, No. XXII, 2001, pp. 3-13.