Abstract
We are concerned with the problem of finding among all polynomials of degreen with leading coefficient 1, the one which has minimal uniform norm on the union of two disjoint compact sets in the complex plane. Our main object here is to present a class of disjoint sets where the best approximation can be determined explicitly for alln. A closely related approximation problem is obtained by considering all polynomials that have degree no larger thann and satisfy an interpolatory constraint. Such problems arise in certain iterative matrix computations. Again, we discuss a class of disjoint compact sets where the optimal polynomial is explicitly known for alln.
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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.
N. I. Achieser (1933):Über einige Funktionen, welche in zwei gegebenen Intervallen am wenigsten von Null abweichen II. Teil. Bull. Acad. Sci. URSS, S. VII,3:309–344.
N. I. Achieser (1933):Über einige Funktionen, welche in zwei gegebenen Intervallen am wenigsten von Null abweichen III, Teil. Bull. Acad. Sci. URSS, S. VII,4:499–536.
N. I. Achieser (1934):Über eine Eigenschaft der “elliptischen” Polynome. Comm. Kharkov Math. Soc.,9:3–8.
L. V. Ahlfors (1979): Complex Analysis, 3rd ed. New York: McGraw-Hill.
R. S. Anderssen, G. H. Goulub (1972):Richardson's non-stationary matrix iterative procedure. Rep. STAN-CS-72-304, Computer Science Deptartment, Stanford University.
B. Atlestam (1977):Tschebycheff-polynomials for sets consisting of two disjoint intervals with application to convergence estimates for the conjugate gradient method. Res. Rep. 77.06 R, Department of Computer Science, Chalmers University of Technology and the University of Göteborg.
P. L. Chebyshev (1899): Oeuvre I. (A. Markoff, N. Sonin, eds.), St. Petersbourg: L'Academie Imperiale de Science.
R. Courant, D. Hilbert (1924): Methoden der Mathematischen Physik I. Berlin: Springer-Verlag.
C. de Boor, J. R. Rice (1982):Extremal polynomials with application to Richardsons iteration for indefinite linear systems. SIAM J. Sci. Statist. Comput.,3:47–57.
M. Eiermann, W. Niethammer, R. S. Varga (1985).A study of semi-iterative methods for nonsymmetric systems of linear equations. Numer. Math.,47:505–533.
G. Faber (1920):Über Tschebyscheffsche Polynome. Crelles J.,150:79–106.
B. Fischer, G. H. Golub (1991):On generating polynomials which are orthogonal over several intervals. Math. Comp.,56:711–730.
J. F. Gracar (1981):Analyses of the Lanczos algorithm and of the approximation problem in Richardson's method. Rept. No UIUCDCS-R-81-1074, Department of Computer Science, University of Illinois at Urbana-Champaign.
P. Henrici (1986): Applied and Computational Complex Analysis, Vol. 3. New York: Wiley.
H. Kober (1957): Dictionary of Conformal Representations. New York: Dover.
E. I. Krupickii (1961):On a class of polynomials deviating least from zero on two intervals. Soviet Mathe.,2:657–660.
V. I. Lebedev (1969):Iterative methods for solving opertor equations with a spectrum contained in several intervals. U.S.S.R. Comput. Math. and Math. Phys.,9:17–24.
W. Markoff (1916):Über Polynome, die in einem gegebenen Intervalle möglichst wenig von Null abweichen. Math. Ann.,77:213–258.
F. Peherstorfer (1984):Extremalpolynome in der L 1-und L 2-Norm auf Zwei disjunkten Intervallen. In: Approximation Theory and Functional Analysis (P. L. Butzer, R. L. Stens, B. Sz.-Nagy, eds.), ISNM.65. Basel: Birkhäuser, 269–280.
F. Peherstorfer (1985):On Tchebycheff polynomials on disjoint intervals. 49. Haar Mem. Conf., Budapest: Colloq. Math. Soc. BOLYAI, 737–751.
F. Peherstorfer (1988):Orthogonal polynomials in L 1-approximation. J. Approx. Theory,52:241–268.
F. Peherstorfer (1990):Orthogonal-and Chebyshev polynomials on two intervals, Acta Math. Hungar.,55:245–278.
T. J. Rivlin (1974): The Chebyshev Polynomials. New York: Wiley.
T. J. Rivlin (1980):Best uniform approximation 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: Constructive Theory. Cambridge, Mass.: M.I.T. Press.
L. N. Trefethen (1981):Rational Chebyshev approximation on the unit disk. Numer. Math.,37:297–320.
J. L. Walsh (1960): Interpolation and Approximation by Rational Functions in the Complex Domain, 3rd ed. Providence, R.I.: American Mathematical Society.
E. T. Whittaker, G. N. Watson (1965): A Course of Modern Analysis, 4th ed. reprinted. London: Cambridge University Press.
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Communicated by Doron S. Lubinsky
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Fischer, B. Chebyshev polynomials for disjoint compact sets. Constr. Approx 8, 309–329 (1992). https://doi.org/10.1007/BF01279022
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DOI: https://doi.org/10.1007/BF01279022