  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.
