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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

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