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On Explicit Decomposition for Positive Polynomials on [-1, +1] with Applications to Extremal Problems

Published online by Cambridge University Press:  20 November 2018

R. Pierre
R. Pierre*
Affiliation:
Université Laval, Québec, Québec
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The following well known inequality was first proved by Bernstein [2].

THEOREM A. If pn(x) is a polynomial of degree n, such that |pn(x)| ≦ 1 for –1 ≦ x = +1, then

1

The dominant n(1 – x2)–;1/2 is best possible only at the zeros of the Tchebychev polynomial

but the bound is precise at every interior point as far as the exponent of n is concerned.

Theorem A was extended to the case of higher derivatives by Duffin and Schaeffer in [4]. In that paper they make extensive use of the oscillation property of the polynomial Tn(x) and of the related function

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 36 , Issue 6 , 01 December 1984 , pp. 1031 - 1045
Copyright
Copyright © Canadian Mathematical Society 1984

References

1. Boas, R. P. Jr., Entire functions (Academic Press, 1954).Google Scholar
2. Bernstein, S., Sur l'ordre de la meilleure approximation des fonctions continues par des polynômes de degré donné, Mémoire de l'Académie Royale de Belgique 4 (1912), 1–104.Google Scholar
3. Bernstein, S., Estimates of derivatives of polynomials, Collected papers, 1, paper n° 46 (1952), 497–499.Google Scholar
4. Duffin, R. J. and Schaeffer, A. C., On some inequalities of S. Bernstein and W. Markov for derivatives of polynomials, Bull. Am. Math. Soc. 44 (1938), 289–297.Google Scholar
5. Karlin, S. and Shapley, L. S., Geometry of moment spaces, Mem. Am. Math. 12 (1953).Google Scholar
6. Markov, W., Über Polynome, die in einem gegebenen Intervalle moglichst wenig von Null abweichen, Math. Ann. 77 (1916), 218–258.Google Scholar
7. Pierre, R. and Rahman, Q. I., On a problem of Turan about polynomials III, Can. J. Math. 34 (1982), 888–889.Google Scholar
8. Szegö, G., Orthogonal polynomials, American Mathematical Society Colloquium Publications 23. Published by the American Mathematical Society. (Providence, Rhode Island, Third edition, 1967).Google Scholar
9. Videnskii, V. S., Generalization of a theorem of A. A. Markov on the estimation of the derivative of a polynomial, Dokl. Akad. Nauk. SSSR 125 (1959), 15–18.Google Scholar
10. Videnskii, V. S., Generalization of the inequalities of V. A. Markov, Dokl. Akad. Nauk. SSSR 120 (1958), 447–450.Google Scholar