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A note on some peculiar nonlinear extremal phenomena of the Chebyshev polynomials

Published online by Cambridge University Press:  20 January 2009

Holger Dette
Holger Dette
Affiliation:
Institut für Mathematische Stochastik, Abteilung Mathematik, Technische Universität Dresden, Mommsenstr, 13, 01062 Dresden, Germany
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Abstract

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We consider the problem of maximizing the sum of squares of the leading coefficients of polynomials (where Pj(x) is a polynomial of degree j) under the restriction that the sup-norm of is bounded on the interval [ −b, b] (b>0). A complete solution of the problem is presented using duality theory of convex analysis and the theory of canonical moments. It turns out, that contrary to many other extremal problems the structure of the solution will depend heavily on the size of the interval [ −b, b].

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 38 , Issue 2 , June 1995 , pp. 343 - 355
Copyright
Copyright © Edinburgh Mathematical Society 1995

References

REFERENCES

1.Achieser, N. I., Theory of Approximation (Dover, New York, 1956).Google Scholar
2.Chihara, T. S., An Introduction to Orthogonal Polynomials (Gordon and Breach, New York, 1978).Google Scholar
3.Dette, H., Optimal designs for identifying the degree of a polynomial regression, Ann. Statist. (1995), to appear.Google Scholar
4.Dette, H., Extremal properties for ultraspherical polynomials, J. Approx. Theory 76 (1994), 246–273.CrossRefGoogle Scholar
5.Dette, H., New identities for orthogonal polynomials on a compact interval, J. Math. Anal. Appl. 179 (1994), 547–573.Google Scholar
6.Lau, T. S.Studden, W. J., On an extremal problem of Fejér, J. Approx. Theory 53 (1988), 184–194.CrossRefGoogle Scholar
7.Natanson, I. P., Konstruktive Funktionentheorie (Akademie Verlag, Berlin, 1955).Google Scholar
8.Perron, O., Die Lehre von den Kettenbrüchen (Band I, II) (B. G. Teubner, Stuttgart, 1954).Google Scholar
9.Pukelsheim, F., Optimal Design of Experiments (Wiley, New York, 1993).Google Scholar
10.Rivlin, T. J., Chebyshev polynomials (Wiley, New York, 1990).Google Scholar
11.Studden, W. J., D s-optimal designs for polynomial regression using continued fractions, Ann. Statist. 8 (1980), 1132–1141.CrossRefGoogle Scholar
12.Studden, W. J., On a problem of Chebyshev, J. Approx. Theory 29 (1981), 253–260.CrossRefGoogle Scholar
13.Wall, H. S., Analytic theory of continued fractions (Van Nostrand, New York, 1948).Google Scholar