Open Access
2011 Maximality of the Bernstein polynomials
Christopher Frayer, Christopher Shafhauser
Involve 4(4): 307-315 (2011). DOI: 10.2140/involve.2011.4.307


For fixed a and b, let Qn be the family of polynomials q(x) all of whose roots are real numbers in [a,b] (possibly repeated), and such that q(a)=q(b)=0. Since an element of Qn is completely determined by it roots (with multiplicity), we may ask how the polynomial is sensitive to changes in the location of its roots. It has been shown that one of the Bernstein polynomials bi(x)=(xa)ni(xb)i, i=1,,n1, is the member of Qn with largest supremum norm in [a,b]. Here we show that for p1, b1(x) and bn1(x) are the members of Qn that maximize the Lp norm in [a,b]. We then find the associated maximum values.


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Christopher Frayer. Christopher Shafhauser. "Maximality of the Bernstein polynomials." Involve 4 (4) 307 - 315, 2011.


Received: 5 May 2010; Revised: 4 May 2011; Accepted: 12 July 2011; Published: 2011
First available in Project Euclid: 20 December 2017

zbMATH: 1244.26027
MathSciNet: MR2905232
Digital Object Identifier: 10.2140/involve.2011.4.307

Primary: 30C15

Keywords: $L^p$ norm , Bernstein polynomial , polynomial root dragging

Rights: Copyright © 2011 Mathematical Sciences Publishers

Vol.4 • No. 4 • 2011
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