Vol. 4, No. 4, 2011

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Maximality of the Bernstein polynomials

Christopher Frayer and Christopher Shafhauser

Vol. 4 (2011), No. 4, 307–315

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) = (x a)ni(x b)i, i = 1,,n 1, is the member of Qn with largest supremum norm in [a,b]. Here we show that for p 1, 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.

polynomial root dragging, $L^p$ norm, Bernstein polynomial
Mathematical Subject Classification 2000
Primary: 30C15
Received: 5 May 2010
Revised: 4 May 2011
Accepted: 12 July 2011
Published: 21 March 2012

Communicated by Martin Bohner
Christopher Frayer
Mathematics Department
University of Wisconsin-Platteville
1 University Plaza
Platteville, WI 53818
United States
Christopher Shafhauser
Department of Mathematics
University of Nebraska-Lincoln
Lincoln, NE 68588
United States