Maximality of the Bernstein polynomials

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.