2 research outputs found
Strong equivalences of approximation numbers and tractability of weighted anisotropic Sobolev embeddings
In this paper, we study multivariate approximation defined over weighted
anisotropic Sobolev spaces which depend on two sequences and of positive numbers. We
obtain strong equivalences of the approximation numbers, and necessary and
sufficient conditions on , to achieve various notions of
tractability of the weighted anisotropic Sobolev embeddings.Comment: 20 page
How anisotropic mixed smoothness affects the decay of singular numbers of Sobolev embeddings
We continue the research on the asymptotic and preasymptotic decay of
singular numbers for tensor product Hilbert-Sobolev type embeddings in high
dimensions with special emphasis on the influence of the underlying dimension
. The main focus in this paper lies on tensor products involving univariate
Sobolev type spaces with different smoothness. We study the embeddings into
and . In other words, we investigate the worst-case approximation
error measured in and when only linear samples of the function
are available. Recent progress in the field shows that accurate bounds on the
singular numbers are essential for recovery bounds using only function values.
The asymptotic bounds in our setting are known for a long time. In this paper
we contribute the correct asymptotic constant and explicit bounds in the
preasymptotic range for . We complement and improve on several results in
the literature. In addition, we refine the error bounds coming from the setting
where the smoothness vector is moderately increasing, which has been already
studied by Papageorgiou and Wo{\'z}niakowski