  For a function $f$ that is piecewise analytic on a quasi-smooth arc
$\mathcal{L}$ and any $0<\sigma<1$ we construct a sequence of "near-best"
polynomials that converge at a rate $e^{-n^{\sigma}}$ at each point of
analyticity of $f$ and are close to the best polynomial approximants on the
whole $\mathcal{L}$. Also we give examples of quasi-smooth arcs for which
convergence of "near-best" approximants at points of analiticity of $f$ is
geometric, i.e. has a rate $e^{-cn}$ with some $c>0$.
