Entropy and approximation numbers: Singularities and optimality

• Applicant: Dr. Therese Mieth, Ph.D.
• Subject Area: Mathematics
• Term: from 2017 to 2018
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 365818999

Entropy and approximation numbers are well-established concepts that serve in many situations as an effective measure for compactness in Hilbert and non-Hilbert space settings e.g. to capture phenomena from spectral theory, approximation theory, operator theory or learning theory. However, although many authors dealt with the investigation of their asymptotic behaviour it was a frequent practice to involve only unspecific constants in corresponding estimates, neglecting the dependence of certain parameters like the dimension of the underlying domain or the chosen norm. In this regard, this project aims at extending (preasymptotic) estimates for the $L_p$-approximation of periodic and non-periodic Sobolev functions with a particular emphasis on the hidden structure of optimal constants. A further goal is to examine the exact decay order of constants related to entropy numbers of the unit ball of Gaussian reproducing kernel Hilbert spaces. Apart from this, this project investigates entropy and approximation numbers of classical Sobolev embeddings where singularities related to the structure of the underling domain or to the behaviour of inserted weights may arise. Therefore, we plan to use a bracketing technique which extends the method of Dirichlet-Neumann-bracketing from spectral $L_2$-theory to the general case of Banach spaces. In particular, we intend to establish necessary and sufficient conditions on the geometry of unbounded domains that ensure compactness of classical Sobolev embeddings into $L_p$ and to determine how the properties of the domain influence the degree of compactness.

• DFG Programme: Research Grants
• Co-Investigator: Professor Dr. Thomas Kühn