Geodesic
On Explicit Decomposition for Positive Polynomials on [-1, +1] with Applications to Extremal Problems
Canadian journal of mathematics, Tome 36 (1984) no. 6, pp. 1031-1045

Voir la notice de l'article provenant de la source Cambridge University Press

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 – x 2)–;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 
Pierre, R. On Explicit Decomposition for Positive Polynomials on [-1, +1] with Applications to Extremal Problems. Canadian journal of mathematics, Tome 36 (1984) no. 6, pp. 1031-1045. doi: 10.4153/CJM-1984-059-3
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[1] 1. Boas, R. P. Jr., Entire functions (Academic Press, 1954). Google Scholar

[2] 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] 3. Bernstein, S., Estimates of derivatives of polynomials, Collected papers, 1, paper n° 46 (1952), 497–499. Google Scholar

[4] 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] 5. Karlin, S. and Shapley, L. S., Geometry of moment spaces, Mem. Am. Math. 12 (1953). Google Scholar

[6] 6. Markov, W., Über Polynome, die in einem gegebenen Intervalle moglichst wenig von Null abweichen, Math. Ann. 77 (1916), 218–258. Google Scholar

[7] 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] 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] 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] 10. Videnskii, V. S., Generalization of the inequalities of V. A. Markov, Dokl. Akad. Nauk. SSSR 120 (1958), 447–450. Google Scholar

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