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Chebyshev Polynomials

Required data

A univariate Polynomial whose SOSINEQ range specification is desired

The Problem

Let $p_n(x)$ be a univariate polynomial of degree $n,with\gamma$ being the coefficient of $x^n$.

$\underset{\gamma }{\max }\gamma$, subject to:
$|p_n(x)|\le 1,\mathrm{\forall }x\in [-1,1]$.

Formulation

The absolute value constraint can be easily rewritten using two inequalities, namely:

$\begin{array}{c}1+p_n(x)\ge 0\\ 1-p_n(x)\ge 0,\\ \mathrm{\forall }x\in [-1,1]\end{array}$.

Solution

The optimal solution is ${\gamma }^{*}=2^{n-1}$, with $p_n^{*}(x)=arccos(cosnx)$ being the n-th Chebyshev polynomial of the first kind.

Example

Code example in SOStools probably worthwhile!

Reference

SOS tools manual: https://sums-of-squares.github.io/sos/

Related LMIs

SOS Basics

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