On regular matrices that induce the Gibbs phenomenon

Let $s = \{ {s_n}(z)\}$ be a sequence of complex valued functions defined in a subset $D$ of the complex plane and suppose that ${s_n}(z)$ converges to $f(z)$ for $z \in D$. For ${z_0} \in \overline D$ let $K({z_0};\;s)$ and $K({z_0}; f)$ be the cores of $s$ and $f$ respectively. We say that $s$ does not have the Gibbs phenomenon at ${z_0}$ if $K({z_0}; s) \subseteq K({z_0}; f)$. The regular matrix $A$ is said to induce the Gibbs phenomenon in $s$ if $K({z_0}; s) \subseteq K({z_0}; f)$ but $K({z_0}; As) \nsubseteq K({z_0}; f)$. We characterize those regular matrices that induce the Gibbs phenomenon.