The generalized Hadamard product of polynomials and its stability

For two polynomials of degrees $n$ and $m$ ($n\ge m$) $$f\left(s\right)=a_0+a_{1}s+\dots +a_{n-1}{s}^{n-1}+a_n{s}^n$$ and $$g\left(s\right)=b_0+b_{1}s+\dots +b_{m-1}{s}^{m-1}+b_m{s}^m$$ we define a set of polynomials $f\bullet g=\{F_0,\dots ,F_{n-m}\}$, where $$F_j\left(s\right)=a_j{b}_0+a_{j+1}{b}_{1}s+\dots +a_{j+m}{b}_m{s}^m,$$ for $j=0,\dots ,n-m$, 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.