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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 $x^{n}$.

$\max _{\gamma }\gamma$, subject to:
$|p_{n}(x)|\leq 1,\forall x\in [-1,1]$.

Formulation

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

${\begin{aligned}1+p_{n}(x)\geq 0\\1-p_{n}(x)\geq 0,\\\forall x\in [-1,1]\end{aligned}}$.

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