#### 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
##### Abstract

For fixed $a$ and $b$, let ${Q}_{n}$ be the family of polynomials $q\left(x\right)$ all of whose roots are real numbers in $\left[a,b\right]$ (possibly repeated), and such that $q\left(a\right)=q\left(b\right)=0$. 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 ${b}_{i}\left(x\right)={\left(x-a\right)}^{n-i}{\left(x-b\right)}^{i}$, $i=1,\dots ,n-1$, is the member of ${Q}_{n}$ with largest supremum norm in $\left[a,b\right]$. Here we show that for $p\ge 1$, ${b}_{1}\left(x\right)$ and ${b}_{n-1}\left(x\right)$ are the members of ${Q}_{n}$ that maximize the ${L}^{p}$ norm in $\left[a,b\right]$. We then find the associated maximum values.

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