Sharp estimates for the covering numbers of the Weierstrass fractal kernel

TL;DR

This paper introduces a Weierstrass fractal kernel, characterizes its associated RKHS, and provides sharp estimates for the covering numbers of its unit ball, revealing the space's dense subset of nowhere differentiable functions.

Contribution

It offers the first explicit characterization of the RKHS for a Weierstrass fractal kernel and derives sharp covering number estimates for its unit ball.

Findings

RKHS contains a dense subset of continuous nowhere differentiable functions

Sharp estimates for covering numbers of the RKHS unit ball are provided

The space's properties highlight its fractal and complex structure

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 the properties of the reproducing kernel Hilbert space (RKHS) associated with 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.