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On Polynomials with Curved Majorants

Published online by Cambridge University Press:  20 November 2018

D. J. Newman  and
T. J. Rivlin
D. J. Newman
Affiliation:
Temple University, Philadephia, Pennsylvania
T. J. Rivlin
Affiliation:
Watson Research Center, IBM, Yorktown Heights, New York
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A well-known result of Chebyshev is that if pn ∊ Pn , (Pn is the set of polynomials of degree at most n) and

(1)

then an(pn), the leading coefficient of pn , satisfies

(2)

with equality holding only for pn = ±Tn , where Tn is the Chebyshev polynomial of degree n. (See [6, p. 57].) This is an example of an extremal problem in which the norm of a given linear operator on Pn is sought. Another example is A. A. Markov's result that (1) implies that

(3)

There are also results for the linear functionals pn(k) (x 0), x 0 real, k = 1, … n – 1 ([8]).

Suppose φ(x) ≧ 0 on [–1, 1] and (1) is generalized to

as suggested by Rahman [4] (polynomials with curved majorants), what can then be said about the analogue of (3) or similar extremal problems?

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 34 , Issue 4 , 01 August 1982 , pp. 961 - 968
Copyright
Copyright © Canadian Mathematical Society 1982

References

References>

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