1. Introduction 2. Order and type in classical complex analysis 3. Relative order and type of convex functions 4. Order and type in duality 5. The infimal convolution 6. The order of an entire function 7. A geometric characterisation of the relative order 8. An extension problem for holomorphic functions 9. Whittaker's decomposition theorem 10. Inequalities for the infimal convolution 11. Inequ...

The infimal convolution of M-convex functions is M-convex. This is a fundamental fact in discrete convex analysis that is often useful in its application to mathematical economics and game theory. M-convexity and its variant called M-convexity are closely related to gross substitutability, and the infimal convolution operation corresponds to an aggregation. This note provides a succinct descrip...

Recently infimal convolution type functions were used in regularization terms of variational models for restoring and decomposing images. This is the first attempt to generalize the infimal convolution of first and second order differences to manifold-valued images. We propose both an extrinsic and an intrinsic approach. Our focus is on the second one since the summands arising in the infimal c...

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 st...

As first demonstrated by Chambolle and Lions the staircasing effect of the RudinOsher-Fatemi model can be reduced by using infimal convolutions of functionals containing higher order derivatives. In this paper, we examine a modification of such infimal convolutions in a general discrete setting. For the special case of finite difference matrices, we show the relation of our approach to the cont...

In this paper, we give a fundamental convexity preserving for spectral functions. Indeed, we investigate infimal convolution, Moreau envelope and proximal average for convex spectral functions, and show that this properties are inherited from the properties of its corresponding convex function. This results have many applications in Applied Mathematics such as semi-definite programmings and eng...