The generalized Hadamard product of polynomials and its stability

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.