Parity of Polynomial Multiplier Sequences for the Chebyshev Basis

We demonstrate that if $p\in \mathbb{R}[x]$ and $p$ is not an even function, then $\{p(k){\}}_{k=0}^{\mathrm{\infty }}$ 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.