Local behaviour of polynomials

Dryanov, D.; Qazi, M.; Rahman, Q.

doi:10.1090/S0025-5718-03-01585-0

Local behaviour of polynomials
HTML articles powered by AMS MathViewer

by D. P. Dryanov, M. A. Qazi and Q. I. Rahman PDF

Math. Comp. 73 (2004), 1345-1364 Request permission

Abstract:

In this paper we study the local behaviour of a trigonometric polynomial $t(\theta ) := \sum _{\nu =-n}^{n} a_{\nu } e^{\mathrm {i}\nu \theta }$ 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

N. I. Achieser, Theory of approximation, Frederick Ungar Publishing Co., New York, 1956. Translated by Charles J. Hyman. MR 0095369
S.N. Bernstein, Sur une propriété des polynômes, Comm. Soc. Math. Kharkow Sér. 2 14 (1913), 1–6.
Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
R. P. Boas Jr., Inequalities for polynomials with a prescribed zero, Studies in mathematical analysis and related topics, Stanford Univ. Press, Stanford, Calif., 1962, pp. 42–47. MR 0150269
Garrett Birkhoff and Morgan Ward, A characterization of Boolean algebras, Ann. of Math. (2) 40 (1939), 609–610. MR 9, DOI 10.2307/1968945
Saunders MacLane, Steinitz field towers for modular fields, Trans. Amer. Math. Soc. 46 (1939), 23–45. MR 17, DOI 10.1090/S0002-9947-1939-0000017-3
G. Pólya and G. Szegő, Problems and theorems in analysis. Vol. II, Revised and enlarged translation by C. E. Billigheimer of the fourth German edition, Springer Study Edition, Springer-Verlag, New York-Heidelberg, 1976. Theory of functions, zeros, polynomials, determinants, number theory, geometry. MR 0465631
M. Riesz, Formule d’interpolation pour la dérivée d’un polynôme, C. R. Acad. Sci. Paris 158 (1914), 1152–1154.
M. Riesz, Eine trigonometrische Interpolationsformel und einige Ungleichungen für Polynome, Jber. Deutsch. Math. Verein. 23 (1914), 354–368.
I. Schur, Über das Maximum des absoluten Betrages eines Polynoms in einem gegebenen Intervall, Math. Z. 4 (1919), 271–287.
A. R. Collar, On the reciprocation of certain matrices, Proc. Roy. Soc. Edinburgh 59 (1939), 195–206. MR 8