Publication: Differentiable approximation and extension of convex functions
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2019-05-10
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Universidad Complutense de Madrid
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Abstract
Las funciones convexas y, en particular, las funciones convexas diferenciables juegan un papel fundamental dentro del Análisis Matemático y tienen una gran cantidad de aplicaciones en otras disciplinas como la Geometría Diferencial, la teoría de EDP’S (por ejemplo, las ecuaciones de Monge-Ampère), las Dinámicas No Nineales, la Computación Cuántica o la Economía. Por tanto, es sin duda útil cualquier herramienta que permita aproximar o extender por funciones convexas diferenciables en distintos espacios de Banach. Si tenemos una función convexa y acotada en conjuntos acotados definida en un espacio de Banach con dual LUR, se sabe por los resultados de D. Azagra que esta función puede aproximarse por funciones convexas diferenciables uniformemente en todo el espacio. Sin embargo, puesto que existen ejemplos defunciones convexas continuas no acotadas en conjuntos acotados, es deseable eliminar esta restricción de la función que deseamos aproximar. En esta tesis se elimina esta restricción y se demuestra que las funciones convexas y continuas definidas en abiertos convexos de espacios de Banach con dual LUR, pueden aproximarse uniformemente por funciones convexas diferenciables uniformemente en todo el espacio. Esto es consecuencia de un resultado más general que muestra que el problema de aproximación uniforme de funciones convexas por funciones convexas de una cierta clase de diferenciabilidad puede reducirse al caso en el que las funciones a aproximar son, además, Lipschitzianas...
The class of convex functions and, in particular, the class of differentiable convex functions play a very important role in the field of Mathematical Analysis and they have plenty of applications in other disciplines such as Differential Geometry, PDE theory (for instance, Monge-Ampère equations), Non linear Dynamics, Quantum Computing or Economics. Therefore, it is no doubt useful to be able to approximate or to extend by differentiable convex functions in various Banach spaces. If we are given a convex function bounded on bounded sets defined on a Banach space whose dual norm is LUR, recent results by D. Azagra ensure that this function can be approximated by differentiable convex functions uniformly on the whole space. However, since there are example of continuous convex functions which are not bounded on bounded subsets, it is desirable to improve these results in such away that this restriction on the function to be approximated can be removed. In this thesis, we drop this assumption and show that every continuous convex function (not necessarily bounded on bounded subsets) defined on an open convex subset of a Banach space whose dual norm is LUR, can be uniformly approximated by differentiable convex functions. This result is a consequence of a more general result which shows that the problem of approximating continuous convex functions uniformly by convex functions of a certain differentiability class can be reduced to the case when the original functions are, in addition, Lipschitz...
The class of convex functions and, in particular, the class of differentiable convex functions play a very important role in the field of Mathematical Analysis and they have plenty of applications in other disciplines such as Differential Geometry, PDE theory (for instance, Monge-Ampère equations), Non linear Dynamics, Quantum Computing or Economics. Therefore, it is no doubt useful to be able to approximate or to extend by differentiable convex functions in various Banach spaces. If we are given a convex function bounded on bounded sets defined on a Banach space whose dual norm is LUR, recent results by D. Azagra ensure that this function can be approximated by differentiable convex functions uniformly on the whole space. However, since there are example of continuous convex functions which are not bounded on bounded subsets, it is desirable to improve these results in such away that this restriction on the function to be approximated can be removed. In this thesis, we drop this assumption and show that every continuous convex function (not necessarily bounded on bounded subsets) defined on an open convex subset of a Banach space whose dual norm is LUR, can be uniformly approximated by differentiable convex functions. This result is a consequence of a more general result which shows that the problem of approximating continuous convex functions uniformly by convex functions of a certain differentiability class can be reduced to the case when the original functions are, in addition, Lipschitz...
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Tesis de la Universidad Complutense de Madrid, Facultad de Ciencias Matemáticas, Departamento de Análisis Matemáticos, leída el 07-09-2018