ε-Coverings of Hölder-Zygmund Type Spaces on Data-Defined Manifolds

Abstract

We first determine the asymptotes of the $\epsilon$-covering numbers of Hölder-Zygmund type spaces on data-defined manifolds. Secondly, a fully discrete and finite algorithmic scheme is developed providing explicit $\epsilon$-coverings whose cardinality is asymptotically near the $\epsilon$-covering number. Given an arbitrary Hölder-Zygmund type function, the nearby center of a ball in the $\epsilon$-covering can also be computed in a discrete finite fashion.