In the first part of this lecture we study the concept of entropy numbers. A quantity giving a refined quantification of the compactness of compact linear operators. A breakthrough in the beginning 80's was the proof of Carl's inequality. An inequality that allows to estimate eigenvalues of the corresponding operators by entropy numbers. We apply this theory to some special elliptic operators. In the second part we consider Gelfand, Kolmogorov and approximation numbers. Quantities that are proven objects in classical approximation theory that got new importance in the just recent field of compressed sensing. The third part of this lecture will be about non-linear approximation. We will study the concept of approximation spaces according to Pietsch. The final goal will be to connect this theory to wavelets and discuss consequences for non-linear wavelet approximation in function-spaces.
Depending on the students this course will be held in Englisch or German language.
Der Termin für die Übung wird gemeinsam gefunden. |
