Maximality of the Bernstein polynomials

Frayer, Christopher; Shafhauser, Christopher

doi:10.2140/involve.2011.4.307

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Maximality of the Bernstein polynomials

Christopher Frayer and Christopher Shafhauser

Vol. 4 (2011), No. 4, 307–315

DOI: 10.2140/involve.2011.4.307

Abstract

For fixed $a$ and $b$ , let $Q_{n}$ 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 $Q_{n}$ 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 $b_{i} (x) = {(x - a)}^{n - i} {(x - b)}^{i}$ , $i = 1, \dots, n - 1$ , is the member of $Q_{n}$ with largest supremum norm in $[a, b]$ . Here we show that for $p \geq 1$ , $b_{1} (x)$ and $b_{n - 1} (x)$ are the members of $Q_{n}$ that maximize the $L^{p}$ norm in $[a, b]$ . We then find the associated maximum values.

Keywords

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

Mathematical Subject Classification 2000

Primary: 30C15

Milestones

Received: 5 May 2010

Revised: 4 May 2011

Accepted: 12 July 2011

Published: 21 March 2012

Communicated by Martin Bohner

Authors

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

Vol. 4, No. 4, 2011