Snake polynomials with negative Chebyshev coefficients

Samuel Paik-Heintz

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Abstract

We disprove a conjecture of G. Nikolov that every continuous, even, convex majorant on $[-1,1]$ has snake polynomials with nonnegative Chebyshev expansions. The polynomial $1+\tfrac12x^2-\tfrac1{12}x^4$ has a degree-$4$ snake with a negative coefficient; so does the corner majorant $1+|x|$. For majorants $\mu=\sqrt{R}$, where $R\in\mathbb{R}[x]$ is positive on $[-1,1]$, the snakes form a Bernstein–Szegő family. When $n\ge\deg R$, their Chebyshev coefficients equal the coefficients of the maximal Fejér–Riesz factor of $R$; a reflected formula holds down to $n=\lceil(\deg R)/2\rceil$. This coefficient criterion is strictly stronger than nonnegativity of the Chebyshev coefficients of $R$. For the polynomial counterexample, applying the criterion to $R=\mu^2$ gives a negative coefficient in every snake of degree $n\ge4$, at an interior position for $n\ge5$.

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