Real Interpolation and Convex Analysis (An Introduction)/
Reell Interpolation och Konvex Analys (Introduktion)
Number of credits: 8 hp
Examiner: Natan Kruglyak
Course literature: 1) S. Kislyakov, N. Kruglyak, Extremal problems in interpolation theory, Whitney-Besicovitch coverings, and singular integrals. Birkhäuser, 2013. 2) I. Ekeland, P. Temam, Convex analysis and variational problems, SIAM, 1999. 3) O. Scherzer, M. Grasmair, H. Grossauer, M. Haltmeire, F. Lenzen, Variational methods in imaging, Springer, 2009
Course contents: Extremal problems in Real Interpolation, K, L, and E- functionals. Riesz sunrize lemma and near optimal decomposition for the E- functional. Rudin-Osher-Fatemi denoising model and the L-functional in real interpolation. Convex lower semicontinuous functions on Banach spaces and their properties. Separation theorems. Baire theorem and continuity of convex functions on Banach spaces. Subdifferentibility of convex functions. Examples and counterexamples. Fenchel transform and its properties. Inverse of Fenchel transform for reflexive and non-reflexive Banach spaces. Image of Fenchel transform, importance of weak* topology. Infimal convolution and its properties, connections with real interpolation. Fenchel transform of infimal convolution. Examples. Fenchel dual of the sum of two lower semicontinuous functions. Attouch-Brezis theorem. Minimization under constrains in Banach spaces and real interpolation.
Organisation: Lectures and seminars.
Examination: Oral presentation of assignments given during the course.
Prerequisites: Standard course on functional analysis.
Last updated: 2015-09-07