Quadratic Hyperbolicity Preservers & Multiplier Sequences

TL;DR

This paper characterizes quadratic hyperbolicity-preserving operators and identifies classes of multiplier sequences, advancing understanding of polynomial stability preservation in differential operators.

Contribution

It provides a complete characterization of quadratic hyperbolicity preservers and introduces new classes of multiplier sequences, extending prior partial results.

Findings

Characterization of all quadratic hyperbolicity-preserving operators

Identification of new classes of P_n-multiplier sequences

Extension of known conditions for zero interlacing in differential operators

Abstract

It is known (see \cite[Br\"and\'en, Lemma 2.7]{B10}) that a necessary condition for $T:=\sum Q_k(x) D^k$ to be hyperbolicity preserving is that $Q_k(x)$ and $Q_{k-1}(x)$ have interlacing zeros. We characterize all quadratic linear operators, as a consequence we find several classes of $P_n$-multiplier sequence.