A nonzero boundary witness and failure of regular convergence

Choose a real even compact mollifier $\rho$ of integral one. Extend the compact mixed kernel $K_f$ by zero in the developing coordinates and define

For sufficiently small $\epsilon >0$, all supports remain in a common compact subset of $U^2$. Let $B_{f,\epsilon }$ be the resulting actual polynomial. Equivalently, using compatible local extensions, $B_{f,\epsilon }[h,\varphi ]=B_f[{\rho }_{\epsilon }\ast h,{\rho }_{\epsilon }\ast \varphi ]$. Each approximant has smooth compact kernel and belongs to ${\mathcal{P}}_{\mathrm{reg}}$.