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 $c>0$.
Categories: math.CV