Sharp estimates for the covering numbers of the Weierstrass fractal kernel

Abstract

In this paper we use the infamous continuous and nowhere differentiable Weierstrass function as a prototype to define a "Weierstrass fractal kernel". We investigate properties of the reproducing kernel Hilbert space (RKHS) associated to this kernel by presenting an explicit characterization of this space. In particular, we show that this space has a dense subset composed of continuous but nowhere differentiable functions. Moreover, we present sharp estimates for the covering numbers of the unit ball of this space as a subset of the continuous functions.