Interpolation and Approximation of Piecewise Smooth Functions

This paper provides approximation orders for a class of nonlinear interpolation procedures for uniformly sampled univariate data. The interpolation is based on essentially nonoscillatory (ENO) and subcell resolution (SR) reconstruction techniques. These nonlinear techniques aim at reducing significantly the approximation error for functions with isolated singularities and are therefore attractive for applications such as shock computations or image compression. We prove that in the presence of isolated singularities, the approximation order provided by the interpolation procedure is improved by a factor of $h$ relative to the linear methods, where h is the sampling rate. Moreover, for h below a critical value, we recover the optimal approximation order as for uniformly smooth functions.