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

Pelin Özkartepe

doi:10.17776/csj.405512

TR EN

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

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.

Details

Primary Language

English

Subjects

-

Journal Section

Research Article

Authors

Pelin Özkartepe ^*

Publication Date

March 16, 2018

Submission Date

August 1, 2017

Acceptance Date

February 6, 2018

Published in Issue

Year 2018 Volume: 39 Number: 1

DOI

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

IZ

https://izlik.org/JA57CK98SC

Cite

RIS / Bibtex

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

Cited By

On the behaviour of derivative of algebraic polynomials in the regions with cusps

Applied Mathematics in Science and Engineering

https://doi.org/10.1080/27690911.2022.2076844

Нерiвностi типу Маркова–Нiкольського в областях iз внутрiшнiми нульовими кутами в просторi Бергмана

Ukrains’kyi Matematychnyi Zhurnal

https://doi.org/10.37863/umzh.v75i3.7322

Inequalities of the Markov–Nikol’skii Type in Regions with Zero Interior Angles in the Bergman Space

Ukrainian Mathematical Journal

https://doi.org/10.1007/s11253-023-02208-4

On the growth of the modulus of the derivative of algebraic polynomials in bounded and unbounded domains with cusps

Filomat

https://doi.org/10.2298/FIL2327243G

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

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

Öz

Anahtar Kelimeler

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

Abstract

Keywords

References

Details

Primary Language

Subjects

Journal Section

Authors

Publication Date

Submission Date

Acceptance Date

Published in Issue

DOI

IZ

Cite

Cited By

On the behaviour of derivative of algebraic polynomials in the regions with cusps

Нерiвностi типу Маркова–Нiкольського в областях iз внутрiшнiми нульовими кутами в просторi Бергмана

Inequalities of the Markov–Nikol’skii Type in Regions with Zero Interior Angles in the Bergman Space

On the growth of the modulus of the derivative of algebraic polynomials in bounded and unbounded domains with cusps