Optimal Decomposition in Real Interpolation and Duality in Convex Analysis

This thesis is devoted to the study of mathematical properties of exact minimizers for the K–,L–, and E– functionals of the theory of real interpolation. Recently exact minimizers for these functionals have appeared in important results in image processing. In the thesis, we present a geometry of optimal decomposition for L– functional for the couple (`2, X), where space `2 is defined by the standard Euclidean norm ‖ · ‖2 and where X is a Banach space on Rn. The well known ROF denoising model is a special case of an L– functional for the couple ( L2, BV ) where L2 and BV stand for the space of square integrable functions and the space of functions with bounded variation on a rectangular domain respectively. We provide simple proofs and geometrical interpretation of optimal decomposition by following ideas by Yves Meyer who has used a duality approach to characterize optimal decomposition for ROF denoising model. The operation of infimal convolution is a very important and non–trivial tool in functional analysis and is also very well–known within the context of convex analysis. The L–, K– and E– functionals can be regarded as an infimal convolution of two well defined functions but unfortunately tools from convex analysis can not be applied in a straigtforward way in this context of couples of spaces. We have considered infimal convolution on Banach couples and by using a theorem due to Attouch and Brezis, we have established sufficient conditions for an infimal convolution on a given Banach couple to be subdifferentiable, which turns out to be the most important requirement that an infimal convolution would satisfy for a decomposition to be optimal. We have also provided a lemma that we have named Key Lemma, which characterizes optimal decomposition for an infimal convolution in general. The main results concerning mathematical properties of optimal decomposition for L–, K– and E– functionals for the case of general regular Banach couples are presented. We use a duality approach which can be summarized in three steps: First we consider the concerned functional as an infimal convolution and reformulate the infimal convolution at hand as a minimization of a sum of two specific functions on the intersection of the couple. Then we prove that it is subdifferentiable and finally use the characterizaton of its optimal decomposition. We have also investigated how powerful our approach is by applying it to two well–known optimization problems, namely convex and linear programming. As a result we have obtained new proofs for duality theorems which are central for these problems.