# The generalized Hadamard product of polynomials and its stability

## TL;DR

This paper introduces a generalized Hadamard product of polynomials, establishes conditions for its Hurwitz stability, and extends classical stability results, with implications for polynomial factorization and stability analysis.

## Contribution

It defines a new generalized Hadamard product of polynomials, provides sufficient stability conditions, and generalizes the Garloff–Wagner theorem.

## Key findings

- The generalized Hadamard product's stability can be characterized under specific conditions.
- Any polynomial with positive coefficients can be paired with another polynomial to produce a stable product.
- The classical Garloff–Wagner theorem is a special case of the new generalized stability result.

## Abstract

For two polynomials of degrees $n$ and $m$ ($n\geq m$) $$ f\left( s\right) =a_{0}+a_{1}s+\ldots+a_{n-1}s^{n-1}+a_{n}s^{n}$$ and $$g\left( s\right) =b_{0}+b_{1}s+\ldots+b_{m-1}s^{m-1}+b_{m}s^{m}$$ we define a set of polynomials $f\bullet g =\left\{ F_{0},\ldots,F_{n-m}\right\} $, where \[ F_{j}\left( s\right) =a_{j}b_{0}+a_{j+1}b_{1}s+\ldots+a_{j+m}b_{m}s^{m}, \] for $j=0,\ldots,n-m$, and call it \textit{a generalized Hadamard product of $f$ and $g$}. We give sufficient conditions for the Hurwitz stability of $f\bullet g$. The obtained results show that the famous Garloff--Wagner theorem on the Hurwitz stability of the Hadamard product of polynomials is a special case of a more general fact. We also show that for every polynomial with positive coefficients (even not necessarily stable) one can find a polynomial such that their generalized Hadamard product is stable. Some connections with polynomials admitting the Hadamard factorization are also given. Numerical examples complete and illustrate the considerations.