# Extremising the $L_p$-norm of a monic polynomial with roots in a given interval and Hermite interpolation

## by Shayne Waldron

## Abstract:

Let $\Theta$ be a multiset of $n$ points in $[a,b]$, and

$$\omega_\Theta:=\prod_{\theta\in\Theta}(\cdot-\theta).$$

In this paper we investigate the extrema of $\Theta\mapsto\norm{\omega_\Theta}_p
$.
Consequences of the results we obtain include: $L_p$-bounds for Hermite
interpolation, error estimates for Gauss quadrature formul{\ae} with multiple
nodes, and certain quantitative statements about good and best approximation
by polynomials of fixed degree.

**Keywords:**
Hermite interpolation, B-spline, Green's function, Beesack's inequality, Wirtinger
inequality

**Math Review Classification:**
xxx (primary), xxx (secondary)

**Length:**
10 pages

**Comment:**
Written in TeX, contains 1 figure. Supported by the Chebyshev professorship of Carl de Boor. See Project Hermite
for related work.

**Last updated:**
25 March 1996

**Status:**
This article will be reincarnated at some date in the near future (to deal with the derivatives of such polynomials, etc.)

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