On shape-preserving probabilistic wavelet approximators

Abstract

We introduce a general class of shape-preserving wavelet approximating operators (approximators) which transform cumulative distribution functions and densities into functions of the same type. Our operators can be considered as a generalization of the operators introduced by Anastassiou and Yu [1]. Further, we extend the consideration by studying the approximation properties for the whole variety of L_p: -norms, 0<p≤∞. In [1] the case p=∞ is discussed. Using the properties of integral moduli of smoothness, we obtain various approximation rates under no (or minimal) additional assumptions on the functions to be approximated. These assumptions are in terms of the function or its Riesz potential belonging to certain homogeneous Besov, Triebel-Lizorkin, Sobolev spaces, the pace BV_p of functions with bounded Wiener-Young p-variation, etc