Operator Diagonalizations of Multiplier Sequences

TL;DR

This paper explores the diagonalization of hyperbolicity preserving operators with respect to Hermite and Laguerre bases, revealing new formulas and characterizations, but noting limitations for other bases like Legendre.

Contribution

It introduces a new linear operator representation for hyperbolicity preservation and provides novel formulas and characterizations for Hermite and Laguerre multiplier sequences.

Findings

Hermite and Laguerre multiplier sequences can be diagonalized into sums of classical multiplier sequences.

The property does not extend to the Legendre basis.

New formulas for Peetre's $Q_k$'s and algebraic characterizations of Hermite multiplier sequences.

Abstract

We consider hyperbolicity preserving operators with respect to a new linear operator representation on $\mathbb{R}[x]$. In essence, we demonstrate that every Hermite and Laguerre multiplier sequence can be diagonalized into a sum of hyperbolicity preserving operators, where each of the summands forms a classical multiplier sequence. Interestingly, this does not work for other orthogonal bases; for example, this property fails for the Legendre basis. We establish many new formulas concerning the $Q_k$'s of Peetre's 1959 differential representation for linear operators in the specific case of Hermite and Laguerre diagonal differential operators. Additionally, we provide a new algebraic characterization of the Hermite multiplier sequences and also extend a recent result of T. Forg\'acs and A. Piotrowski on hyperbolicity properties of the polynomial coefficients in hyperbolicity preserving Hermite diagonal differential operators.