Polynomial Approximation of Piecewise Analytic Functions on Quasi-Smooth Arcs

For a function f that is piecewise analytic on a quasi-smooth arc L and any 0 < sigma < 1 we construct a sequence of 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 L. Moreover, we give examples when such polynomials can be constructed for sigma = 1.