Ambient Approximation of Functions and Functionals on Embedded Submanifolds

While many problems of approximation theory are already well-understood in Euclidean space and its subdomains, much less is known about problems on submanifolds of that space. And this knowledge is even more limited when the approximation problem presents certain difficulties like sparsity of data samples or noise on function evaluations, both of which can be handled successfully in Euclidean space by minimisers of certain energies. On the other hand, such energies give rise to a considerable amount of techniques for handling various other approximation problems, in particular certain partial differential equations. The present thesis provides a deep going analysis of approximation results on submanifolds and approximate representation of intrinsic functionals: It provides a method to approximate a given function on a submanifold by suitable extension of this function into the ambient space followed by approximation of this extension on the ambient space and restriction of the approximant to the manifold, and it investigates further properties of this approximant. Moreover, a differential calculus for submanifolds via standard calculus on the ambient space is deduced from Riemannian geometry, and various energy functionals are presented and approximately handled by an approximate application of this calculus. This approximate handling of functionals is then employed in several penalty-based methods to solve problems such as interpolation in sparse data sites, smoothing and denoising of function values and approximate solution of certain partial differential equations.

Im Gegensatz zum euklidischen Fall sind auf allgemeinen glatten Untermannigfaltigkeiten viele Probleme der Approximationstheorie noch nicht ausreichend verstanden und gelöst. Die vorliegende Arbeit leistet hier einen Beitrag zum weiteren Verständnis. Sie behandelt insbesondere Probleme der Approximation glatter Funktionen und der Minimierung geeigneter Funktionale. Diese werden außerdem exemplarisch zur Behandlung von Problemen der Datenglättung, der Interpolation und Extrapolation sowie der Lösung elliptischer partieller Differentialgleichungen weiter angewendet.