ABSTRACT

We consider the problem of Ω-weighted Lp-approximation of piecewise r-smooth functions defined in the whole real line R. We assume that the singular points are unknown. The approximations are constructed based on n function values that can be collected adaptively. We give a necessary and sufficient condition for the weight Ω that allows approximations of order n-r . While for globally smooth functions the optimal designs are nonadaptive, for piecewise smooth functions one has to use adaptive sampling to locate and well approximate in vicinity of the singular points. The analysis is done in the worst case and asymptotic settings.