Parity of Polynomial Multiplier Sequences for the Chebyshev Basis

TL;DR

This paper characterizes polynomial multiplier sequences for Chebyshev polynomials, showing non-even polynomials do not form such sequences and providing a complete description of geometric sequences for this basis.

Contribution

It proves that only even polynomials can generate multiplier sequences for Chebyshev polynomials and characterizes all geometric multiplier sequences for this basis.

Findings

Non-even polynomials are not Chebyshev multiplier sequences.

Complete characterization of geometric multiplier sequences.

Provides criteria for polynomial multiplier sequences for 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.