Uniform and Pointwise Polynomial Estimates in Regions with Interior and Exterior Cusps

Pelin Özkartepe

doi:10.17776/csj.405512

Research Article

İç ve Dış Sıfır Açılı Bölgelerde Polinom için Düzgün ve Noktasal Değerlendirmeler

Year 2018, , 47 - 65, 16.03.2018

Pelin Özkartepe

https://doi.org/10.17776/csj.405512

Cited By: 4

Abstract

Bu
çalışmada, içte ve dışta sıfır açı olan parçalı Dini düzgün eğri ile sınırlı
sonlu ve sonsuz bölgelerdeki cebirsel polinomlar için düzgün ve noktasal
değerlendirmeler inceledik.

Keywords

Cebirsel polinomlar

References

[1] Abdullayev F.G., Andrievskii V.V. On the orthogonal polynomials in the domains with -quasiconformal boundary. Izv. Akad. Nauk Azerb. SSR., Ser. FTM 1983; 1: 3-7.
[2] Abdullayev F. G., Özkartepe N. P., Gün C. D. Uniform and pointwise polynomial inequalities in regions without cusps in the weighted Lebesgue space. Bulletin of Tbilisi ICMC , 18-1 (2014) 146-167.
[3] Abdullayev F.G., Özkartepe P. On the growth of algebraic polynomials in the whole complex plane. J. Korean Math. Soc., 52-4 (2015) 699-725.
[4] Abdullayev F. G., Gün C.D., Ozkartepe N.P. Inequalities for algebraic polynomials in regions with exterior cusps. J. Nonlinear Funct. Anal., Article ID 3 (2015) 1-32.
[5] Abdullayev F.G., Özkartepe P. Uniform and pointwise polynomial inequalities in regions with cusps in the weighted Lebesgue space. Jaen Journal on Approximation, 7-2 (2015) 231-261.
[6] Abdullayev F.G., Özkartepe P., Polynomial inequalities in Lavrentiev regions with interior and exterior zero angles in the weighted Lebesgue space. Publications de l'Institut Mathématique (Beograd), 100-114 (2016) 209-227.
[7] Ahlfors L. Lectures on Quasiconformal Mappings. Princeton, NJ: Van Nostrand, 1966.
[8] Andrievskii V.V. On the uniform convergence of the Bieberbach polynomials in the regions with piecewise quasiconformal boundary, In: Theory of Mappings and approximation functions, "Naukovo Dumka", Kyiv, (1983) 3-18. (in Russian)
[9] Andrievskii V.V., Weighted Polynomial Inequalities in the Complex Plane. Journal of Approximation Theory, 164-9 (2012) 1165-1183.
[10] Andrievskii V.V., Belyi V.I. & Dzyadyk V.K. Conformal invariants in constructive theory of functions of complex plane. Atlanta:World Federation Publ.Com., 1995.
[11] Andrievskii V.V., Blatt H.P. Discrepancy of Signed Measures and Polynomial Approximation, Springer Verlag New York Inc., 2010.
[12] Dzjadyk V.K., Introduction to the Theory of Uniform Approximation of Function by Polynomials, Nauka, Moskow, 1977. (in Russian)
[13] Gaier D., On the convergence of the Bieberbach polynomials in regions with corners. Constructive Approximation, 4 (1988) 289-305.
[14] Jackson D., Certain problems on closest approximations. Bull. Amer. Math. Soc., 39 (1933) 889-906.
[15] Lehto O., Virtanen K.I., Quasiconformal Mapping in the plane, Springer Verlag, Berlin, 1973.
[16] Mamedhanov D.I., Inequalities of S.M.Nikol'skii type for polynomials in the complex variable on curves, Soviet Math.Dokl., 15 (1974) 34-37.
[17] Mamedhanov D.I., On Nikol'skii-type inequalities with new characteristics, Doklady Mathematics, 82 (2010) 882-883.
[18] Milovanovic G.V., Mitrinovic D.S. and Rassias Th.M., Topics in Polynomials: Extremal Problems, Inequalities, Zeros, World Scientific, Singapore, 1994.
[19] Nikol'skii S.M., Approximation of function of several variable and imbeding theorems, Springer-Verlag, New-York, 1975.
[20] Özkartepe N. P., Abdullayev F. G. , On the interference of the weight and boundary contour for algebraic polynomials in the weighted Lebesgue spaces I. Ukr. Math. J. 2016; 68(10): (Trans. from Ukr. Mat. Zh., 68-10 (2016) 1365-1379.
[21] Pritsker I.E., Comparing norms of polynomials in one and several variables, Journal of Math. Anal. and Appl., 216 (1997) 685-695.
[22] Pommerenke Ch., Univalent Functions, Göttingen, Vandenhoeck & Ruprecht, 1975.
[23] Rickman S., Characterization of quasiconformal arcs. Ann. Acad. Sci. Fenn., Ser. A, Mathematica, 395 (1966) 30 p.
[24] Stylianopoulos N., Strong asymptotics for Bergman polynomials over domains with corners and applications. Const. Approx., 38 (2013) 59-100.
[25] Suetin P.K., The ordinally comparison of various norms of polynomials in the complex domain, Matematicheskie zapiski Uralskogo Gos. Universiteta, 5-4 (1966) (in Russian).
[26] Suetin P. K., Main properties of the orthogonal polynomials along a circle, Uspekhi Math. Nauk, 21-2,128 (1966) 41-88.
[27] Suetin P.K., On some estimates of the orthogonal polynomials with singularities weight and contour. Sibirskiy Mat. Zhurnal, VIII-3 (1967) 1070-1078 (in Russian).
[28] Szegö G., Zygmund A., On certain mean values of polynomials. Journal d'Analyse Mathematique, 3-1 (1953) 225-244.
[29] Walsh J.L., Interpolation and Approximation by Rational Functions in the Complex Domain, AMS, 1960.

Uniform and Pointwise Polynomial Estimates in Regions with Interior and Exterior Cusps

Year 2018, , 47 - 65, 16.03.2018

Pelin Özkartepe

https://doi.org/10.17776/csj.405512

Cited By: 4

Abstract

In
this work, we investigate the estimation for algebraic polynomials in the
bounded and unbounded regions with piecewise Dini smooth curve having interior
and exterior zero angles.

Keywords

Algebraic polynomials

References

[1] Abdullayev F.G., Andrievskii V.V. On the orthogonal polynomials in the domains with -quasiconformal boundary. Izv. Akad. Nauk Azerb. SSR., Ser. FTM 1983; 1: 3-7.
[2] Abdullayev F. G., Özkartepe N. P., Gün C. D. Uniform and pointwise polynomial inequalities in regions without cusps in the weighted Lebesgue space. Bulletin of Tbilisi ICMC , 18-1 (2014) 146-167.
[3] Abdullayev F.G., Özkartepe P. On the growth of algebraic polynomials in the whole complex plane. J. Korean Math. Soc., 52-4 (2015) 699-725.
[4] Abdullayev F. G., Gün C.D., Ozkartepe N.P. Inequalities for algebraic polynomials in regions with exterior cusps. J. Nonlinear Funct. Anal., Article ID 3 (2015) 1-32.
[5] Abdullayev F.G., Özkartepe P. Uniform and pointwise polynomial inequalities in regions with cusps in the weighted Lebesgue space. Jaen Journal on Approximation, 7-2 (2015) 231-261.
[6] Abdullayev F.G., Özkartepe P., Polynomial inequalities in Lavrentiev regions with interior and exterior zero angles in the weighted Lebesgue space. Publications de l'Institut Mathématique (Beograd), 100-114 (2016) 209-227.
[7] Ahlfors L. Lectures on Quasiconformal Mappings. Princeton, NJ: Van Nostrand, 1966.
[8] Andrievskii V.V. On the uniform convergence of the Bieberbach polynomials in the regions with piecewise quasiconformal boundary, In: Theory of Mappings and approximation functions, "Naukovo Dumka", Kyiv, (1983) 3-18. (in Russian)
[9] Andrievskii V.V., Weighted Polynomial Inequalities in the Complex Plane. Journal of Approximation Theory, 164-9 (2012) 1165-1183.
[10] Andrievskii V.V., Belyi V.I. & Dzyadyk V.K. Conformal invariants in constructive theory of functions of complex plane. Atlanta:World Federation Publ.Com., 1995.
[11] Andrievskii V.V., Blatt H.P. Discrepancy of Signed Measures and Polynomial Approximation, Springer Verlag New York Inc., 2010.
[12] Dzjadyk V.K., Introduction to the Theory of Uniform Approximation of Function by Polynomials, Nauka, Moskow, 1977. (in Russian)
[13] Gaier D., On the convergence of the Bieberbach polynomials in regions with corners. Constructive Approximation, 4 (1988) 289-305.
[14] Jackson D., Certain problems on closest approximations. Bull. Amer. Math. Soc., 39 (1933) 889-906.
[15] Lehto O., Virtanen K.I., Quasiconformal Mapping in the plane, Springer Verlag, Berlin, 1973.
[16] Mamedhanov D.I., Inequalities of S.M.Nikol'skii type for polynomials in the complex variable on curves, Soviet Math.Dokl., 15 (1974) 34-37.
[17] Mamedhanov D.I., On Nikol'skii-type inequalities with new characteristics, Doklady Mathematics, 82 (2010) 882-883.
[18] Milovanovic G.V., Mitrinovic D.S. and Rassias Th.M., Topics in Polynomials: Extremal Problems, Inequalities, Zeros, World Scientific, Singapore, 1994.
[19] Nikol'skii S.M., Approximation of function of several variable and imbeding theorems, Springer-Verlag, New-York, 1975.
[20] Özkartepe N. P., Abdullayev F. G. , On the interference of the weight and boundary contour for algebraic polynomials in the weighted Lebesgue spaces I. Ukr. Math. J. 2016; 68(10): (Trans. from Ukr. Mat. Zh., 68-10 (2016) 1365-1379.
[21] Pritsker I.E., Comparing norms of polynomials in one and several variables, Journal of Math. Anal. and Appl., 216 (1997) 685-695.
[22] Pommerenke Ch., Univalent Functions, Göttingen, Vandenhoeck & Ruprecht, 1975.
[23] Rickman S., Characterization of quasiconformal arcs. Ann. Acad. Sci. Fenn., Ser. A, Mathematica, 395 (1966) 30 p.
[24] Stylianopoulos N., Strong asymptotics for Bergman polynomials over domains with corners and applications. Const. Approx., 38 (2013) 59-100.
[25] Suetin P.K., The ordinally comparison of various norms of polynomials in the complex domain, Matematicheskie zapiski Uralskogo Gos. Universiteta, 5-4 (1966) (in Russian).
[26] Suetin P. K., Main properties of the orthogonal polynomials along a circle, Uspekhi Math. Nauk, 21-2,128 (1966) 41-88.
[27] Suetin P.K., On some estimates of the orthogonal polynomials with singularities weight and contour. Sibirskiy Mat. Zhurnal, VIII-3 (1967) 1070-1078 (in Russian).
[28] Szegö G., Zygmund A., On certain mean values of polynomials. Journal d'Analyse Mathematique, 3-1 (1953) 225-244.
[29] Walsh J.L., Interpolation and Approximation by Rational Functions in the Complex Domain, AMS, 1960.

There are 29 citations in total.

Details

Primary Language	English
Journal Section	Natural Sciences
Authors	Pelin Özkartepe
Publication Date	March 16, 2018
Submission Date	August 1, 2017
Acceptance Date	February 6, 2018
Published in Issue	Year 2018

Cite

APA	Özkartepe, P. (2018). Uniform and Pointwise Polynomial Estimates in Regions with Interior and Exterior Cusps. Cumhuriyet Science Journal, 39(1), 47-65. https://doi.org/10.17776/csj.405512

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