Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

ON THE GROWTH OF ALGEBRAIC POLYNOMIALS IN THE WHOLE COMPLEX PLANE

  • ABDULLAYEV, F.G. (Department of Mathematics Faculty of Arts and Science Mersin University) ;
  • OZKARTEPE, N.P. (Department of Mathematics Faculty of Arts and Science Mersin University)
  • Received : 2014.02.21
  • Published : 2015.06.01
Download PDF
⟨ Previous Next ⟩

Abstract

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

Keywords

References

  1. F. G. Abdullayev, Dissertation, Ph.D., Donetsk, 1986.
  2. F. G. Abdullayev, On some properties of orthogonal polynomials over an area in domains ofthe complex plane. III, Ukrain. Mat. Zh. 53 (2001), no. 12, 1588-1599; translation inUkrainian Math. J. 53 (2001), no. 12, 1934-1948.
  3. F. G. Abdullayev and V. V. Andrievskii, On the orthogonal polynomials in the domains with K-quasiconformal boundary, Izv. Akad. Nauk Azerb. Ser. Fiz.-Tekh. Mat. Nauk 1 (1983), 3-7.
  4. F. G. Abdullayev and P. Ozkartepe, An analogue of the Bernstein-Walsh lemma in Jordan regions of the complex plane, J. Inequal. Appl. 2013 (2013), 7 pp. https://doi.org/10.1186/1029-242X-2013-7
  5. L. Ahlfors, Lectures on Quasiconformal Mappings, Princeton, NJ: Van Nostrand, 1966.
  6. V. V. Andrievskii, Uniform convergence of Bieberbach polynomials in domains with piecewise-quasiconformal boundary, Theory of mappings and approximation of functions, 3-18, "Naukova Dumka", Kiev, 1983.
  7. V. V. Andrievskii, Constructive characterization of harmonic functions in domains with a quasiconformal boundary, Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989), no. 2, 425-438.
  8. V. V. Andrievskii, V. I. Belyi, and V. K. Dzyadyk, Conformal invariants in constructive theory of functions of complex variable, World Federation Publishers Company, Atlanta, GA, 1995.
  9. V. V. Andrievskii and H. P. Blatt, Discrepancy of Signed Measures and Polynomial Approximation, Springer Verlag New York Inc., 2002.
  10. D. Gaier, On the convergence of the Bieberbach polynomials in regions with corners, Constr. Approx. 4 (1988), no. 3, 289-305. https://doi.org/10.1007/BF02075463
  11. E. Hille, G. Szego, and J. D. Tamarkin, On some generalization of a theorem of A. Markoff, Duke Math. J. 3 (1937), no. 4, 729-739. https://doi.org/10.1215/S0012-7094-37-00361-2
  12. D. Jackson, Certain problems of closest approximation, Bull. Amer. Math. Soc. 39 (1933), no. 12, 889-906. https://doi.org/10.1090/S0002-9904-1933-05759-2
  13. O. Lehto and K. I. Virtanen, Quasiconformal Mapping in the Plane, Springer Verlag, Berlin, 1973.
  14. D. I. Mamedhanov, Inequalities of S. M. Nikol'skii type for polynomials of a complex variable on curves, Dokl. Akad. Nauk SSSR 214 (1974), 34-37.
  15. S. N. Mergelyan, Certain questions of the constructive functions of theory, Trudy Mat. Inst. Steklov. 37 (1951), 3-91.
  16. S. M. Nikol'skii, Approximation of Functions of Several Variable and Imbeding Theorems, Springer-Verlag, New-York, 1975.
  17. Ch. Pommerenke, Univalent Functions, Gottingen, Vandenhoeck & Ruprecht, 1975.
  18. I. E. Pritsker, Comparing norms of polynomials in one and several variables, J. Math. Anal. Appl. 216 (1997), no. 2, 685-695. https://doi.org/10.1006/jmaa.1997.5699
  19. S. Rickman, Characterization of quasiconformal arcs, Ann. Acad. Sci. Fenn. Ser. A Math. 395 (1966), 30 pp.
  20. N. Stylianopoulos, Strong asymptotics for Bergman polynomials over domains with corners and applications, Constr. Approx. 38 (2013), no. 1, 59-100. https://doi.org/10.1007/s00365-012-9174-y
  21. P. K. Suetin, Order comparison of various norms of polynomials in a complex region, Ural. Gos. Univ. Mat. Zap. 5 (1966), 91-100.
  22. P. K. Suetin, Some estimates of polynomials, orthogonal on a contour, with peculiarities of weight and contour, Sibirsk. Mat. Z. 8 (1967), 1070-1078.
  23. J. L. Walsh, Interpolation and Approximation by Rational Functions in the Complex Domain, American Mathematical Society, Providence, R.I. 1960.
  24. S. E. Warschawski, Uber das randverhalten der ableitung der ableitung der abbildungsfunktion bei konformer abbildung, Math. Z. 35 (1932), no. 1, 321-456. https://doi.org/10.1007/BF01186562

Cited by

  1. Interference of the Weight and Boundary Contour for Algebraic Polynomials in Weighted Lebesgue Spaces. I vol.68, pp.10, 2017, https://doi.org/10.1007/s11253-017-1313-y
  2. On some properties of the orthogonal polynomials over a contour with general Jacobi weight vol.220, pp.5, 2017, https://doi.org/10.1007/s10958-016-3199-x
  3. On the Sharp Inequalities for Orthonormal Polynomials Along a Contour vol.11, pp.7, 2017, https://doi.org/10.1007/s11785-017-0640-1
  4. Bernstein–Walsh type inequalities in unbounded regions with piecewise asymptotically conformal curve in the weighted Lebesgue space vol.234, pp.1, 2018, https://doi.org/10.1007/s10958-018-3979-6
  5. On the Interference of the Weight and Boundary Contour for Algebraic Polynomials in Weighted Lebesgue Spaces. II vol.69, pp.12, 2018, https://doi.org/10.1007/s11253-018-1479-y