Abstract
LetW N(z)=aNzN+... be a complex polynomial and letT n be the classical Chebyshev polynomial. In this article it is shown that the polynomials (2aN)−n+1Tn(WN), n ∈N, are minimal polynomials on all equipotential lines for {z∈C:|W N(z)|≤1 Λ ImW N(z)=0}
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
N. I. Achieser (1928):Über einige Funktionen, die in gegebenen Intervallen am wenigsten von Null abweichen. Bull. Soc. Phys. Math. Kazan, S. III,3: 1–69.
N. I. Achieser (1932):Über einige Funktionen, welche in zwei gegebenen Intervallen am wenigsten von Null abweichen I. Teil. Bull. Acad. Sci. URSS, S.VII,9:1163–1202.
L. V. Ahlfors (1973): Conformal Invariants—Topics in Geometric Function Theory. New York: McGraw-Hill.
B. Atlestam (1977): Tschebycheff-Polynomials for Sets Consisting of Two Disjoint Intervals with Application to Convergence Estimates for the Conjugate Gradient Method. Res. Rep. 77.06R, Department of Computer Science, Chalmers University of Technology and the University of Göteborg.
B. Fischer (1992):Chebyshev polynomials for disjoint compact sets. Constr. Approx.,8:309–329.
C. F. Gauss (1876): Omnem Functionem Algebraicam rationalem Integram. Gauß Werke Bd. III:1–30.
G. G. Lorentz (1986): Approximation of Functions, 2nd edn., New York: Chelsea.
A. Ostrowski (1933):Über den ersten und vierten Gaußschen Beweis des Fundamentalsatzes der Algebra. Gauß Werke Bd. X 2 Abh. 3:1–18.
A. Ostrowski (1933):Über Nullstellen stetiger Funktionen zweier Variabeln. J. Reine Angew. Math.,170:83–94.
F. Peherstorfer (1985):On Tchebycheff polynomials on disjoint intervals. Haar Mem. Conf., Budapest: Colloquia Mathematica Societatis Janós Bolyai, 49. Amsterdam: North-Holland, pp. 737–751.
F. Peherstorfer (1988):Orthogonal polynomials in L 1-approximation. J. Approx. Theory,52:245–278.
F. Peherstorfer (1990):Orthogonal and Chebyshev polynomials on two intervals. Acta Math. Hungar.,55:245–278.
F. Peherstorfer (1991):On Bernstein-Szegö orthogonal polynomials on several intervals, II. Orthogonal polynomials with periodic recurrence coefficients. J. Approx. Theory,64:123–161.
F. Peherstorfer (1993):Orthogonal and extremal polynomials on several intervals. J. Comput. Appl. Math.,48:187–205.
F. Peherstorfer (1995):Elliptic orthogonal and extremal polynomials: Proc. London Math. Soc.70:605–624.
T.J. Rivlin (1980):Best uniformapproximation by polynomials in the complex plane. In: Approximation Theory III (E. W. Cheney, ed.), New York: Academic Press, pp. 75–86.
V. L. Smirnov, N. A. Lebedev (1968):Functions of a complex variable. London: Iliffe Books.
Author information
Authors and Affiliations
Additional information
Communicated by Stephan Ruscheweyh.
Rights and permissions
About this article
Cite this article
Peherstorfer, F. Minimal polynomials for compact sets of the complex plane. Constr. Approx 12, 481–488 (1996). https://doi.org/10.1007/BF02437504
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02437504