Function space embeddings and problems with infinitely many variables

Abstract

We study a sequence of continuous approximation problems on function spaces, in which the elements depend on a large number of variables; the number of variables is also called the dimension of the problem. In the field of Information Based Complexity, one frequently studies approximation errors when the number of variables tends to infinity, i.e., one studies a whole sequence of function spaces and the corresponding approximation problems. Moreover, one tries to identify situations where a curse of dimensionality can be avoided.

It is known that certain sequences of function spaces with particular features facilitate the error analysis for certain types of algorithms. Embedding theorems for such sequences can help to transfer results from one sequence of function spaces to another. In the existing literature on this subject, reproducing kernel Hilbert spaces with a tensor-product structure are frequently studied. In this talk we outline an approach to deal with embedding results that do not necessarily require the tensor-product structure, but allow for a more general setting.

This talk is based on joint work with Michael Gnewuch (Osnabrueck) and Klaus Ritter (Kaiserslautern).