Parity of Polynomial Multiplier Sequences for the Chebyshev Basis

Abstract: We demonstrate that if $p\in\mathbb{R}[x]$ and $p$ is not an even function, then ${p(k)}^{\infty}_{k=0}$ is not a multiplier sequence for the basis of Chebyshev polynomials of the first kind. We also give a characterization of geometric multiplier sequences for the Chebyshev basis.