Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Local behaviour of polynomials

Authors: D. P. Dryanov, M. A. Qazi and Q. I. Rahman
Journal: Math. Comp. 73 (2004), 1345-1364
MSC (2000): Primary 42A05, 26D05, 26D10, 30C10, 30A10
Published electronically: July 28, 2003
MathSciNet review: 2047090
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we study the local behaviour of a trigonometric polynomial $t(\theta )\,:=\,\sum _{\nu =-n}^{n}\,a_{\nu }\,e^{{i}\nu \theta }$ around any of its zeros in terms of its estimated values at an adequate number of freely chosen points in $[0 \,,\, 2 \pi )$. The freedom in the choice of sample points makes our results particularly convenient for numerical calculations. Analogous results for polynomials of the form $\sum _{\nu =0}^{n}\,a_{\nu }\,x^{\nu }$ are also proved.

References [Enhancements On Off] (What's this?)

  • 1. N.I. Ahiezer, Theory of Approximation, Frederick Ungar Publishing Co., New York, 1956. MR 20:1872
  • 2. S.N. Bernstein, Sur une propriété des polynômes, Comm. Soc. Math. Kharkow Sér. 2 14 (1913), 1-6.
  • 3. R.P. Boas, Jr., Entire Functions, Academic Press, New York, 1954. MR 16:914f
  • 4. R.P. Boas, Jr., Inequalities for polynomials with a prescribed zero, Studies in Mathematical Analysis and related topics (Essays in honour of George Pólya) (Gabor Szegö, ed.), Stanford University Press, Stanford California, 1962, pp. 42-47. MR 27:270
  • 5. P. Erdos, Some remarks on polynomials, Bull. Amer. Math. Soc. 53 (1947), 1169-1176. MR 9:281g
  • 6. C. Hyltén-Cavalius, Some extremal problems for trigonometrical and complex polynomials, Math. Scand. 3 (1955), 5-20. MR 17:247c
  • 7. G. Pólya and G. Szegö, Problems and Theorems in Analysis II, Springer-Verlag, Berlin, Heidelberg, 1976. MR 57:5529
  • 8. M. Riesz, Formule d'interpolation pour la dérivée d'un polynôme, C. R. Acad. Sci. Paris 158 (1914), 1152-1154.
  • 9. M. Riesz, Eine trigonometrische Interpolationsformel und einige Ungleichungen für Polynome, Jber. Deutsch. Math. Verein. 23 (1914), 354-368.
  • 10. I. Schur, Über das Maximum des absoluten Betrages eines Polynoms in einem gegebenen Intervall, Math. Z. 4 (1919), 271-287.
  • 11. P. Turán, On rational polynomials, Acta Scientiarum Mathematicarum (Szeged) 11 (1946), 106-113. MR 8:266c

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 42A05, 26D05, 26D10, 30C10, 30A10

Retrieve articles in all journals with MSC (2000): 42A05, 26D05, 26D10, 30C10, 30A10

Additional Information

D. P. Dryanov
Affiliation: Département de Mathématiques et de Statistique, Université de Montréal, Montréal H3C 3J7, Canada

M. A. Qazi
Affiliation: Department of Mathematics, Tuskegee University, Tuskegee, Alabama 36088

Q. I. Rahman
Affiliation: Département de Mathématiques et de Statistique, Université de Montréal, Montréal H3C 3J7, Canada

Keywords: Trigonometric polynomials, algebraic polynomials, M. Riesz's interpolation formula, Schur's inequality, Bernstein's inequality
Received by editor(s): August 20, 2002
Received by editor(s) in revised form: December 22, 2002
Published electronically: July 28, 2003
Article copyright: © Copyright 2003 American Mathematical Society

American Mathematical Society