Approximation of piecewise Hölder functions from inexact information

Abstract

We study the Lp-approximation of functions f consisting of two smooth pieces separated by an (unknown) singular point sf; each piece is r times differentiable and the rth derivative is Hölder continuous with exponent ϱ. The approximations use n inexact function values yi=f(xi)+ei with |ei|≤δ. Let 1≤p<∞. We show that then the minimal worst case error is proportional to max(δ,n−(r+ϱ)) in the class of functions with uniformly bounded both the Hölder coefficients and the discontinuity jumps |f(sf+)−f(sf−)|. This error is achieved by an algorithm that uses a new adaptive mechanism to approximate sf, where the number of adaptively chosen points xi is only O(lnn). The use of adaption, p<∞, and the uniform bound on the Hölder coefficients and the discontinuity jumps are crucial. If we restrict the class even further to continuous functions, then the same worst case result can be achieved also for p=∞ using no more than (r−1)+ adaptive points. The results generalize earlier results that were obtained for exact information (where δ=0) and f(r) piecewise Lipschitz (where ϱ=1).