Approximating piecewise-smooth functions

We consider the possibility of using locally supported quasi-interpolation operators for the approximation of univariate nonsmooth functions. In such a case, one usually expects the rate of approximation to be lower than that of smooth functions. It is shown in this paper that prior knowledge of the type of 'singularity' of the function can be used to regain the full approximation power of the quasi-interpolation method. The singularity types may include jumps in the derivatives at unknown locations or even singularities of the form (x - s)^α, with unknown s and α. The new approximation strategy includes singularity detection and high-order evaluation of the singularity parameters, such as the above s and α. Using the acquired singularity structure, a correction of the primary quasi-interpolation approximation is computed, yielding the final high-order approximation. The procedure is local, and the method is also applicable to a nonuniform data-point distribution. The paper includes some examples illustrating the high performance of the suggested method, supported by an analysis proving the approximation rates in some of the interesting cases.