Chebyshev polynomials corresponding to a vanishing weight

TL;DR

This paper extends the theory of weighted Chebyshev polynomials on the unit circle to non-integer weights, relating them to lemniscates and generalizing key inequalities.

Contribution

It generalizes Chebyshev polynomial results for weights $(z-1)^s$ to non-integer $s$, linking them to lemniscates and broadening the Erdős–Lax inequality.

Findings

Extended Chebyshev polynomial theory to non-integer weights

Connected Chebyshev polynomials on lemniscates to classical categories

Broadened Erdős–Lax inequality for polynomial powers

Abstract

We consider weighted Chebyshev polynomials on the unit circle corresponding to a weight of the form $(z-1)^s$ where $s>0$. For integer values of $s$ this corresponds to prescribing a zero of the polynomial on the boundary. As such, we extend findings of Lachance, Saff and Varga, to non-integer $s$. Using this generalisation, we are able to relate Chebyshev polynomials on lemniscates and other, more established, categories of Chebyshev polynomials. An essential part of our proof involves the broadening of the Erd\H{o}s--Lax inequality to encompass powers of polynomials. We believe that this particular result holds significance in its own right.