Domingo García

Convex Analysis in Polynomial Spaces with Applications is intended to serve a broad audience of undergraduate and graduate students, junior and senior researchers, and as a general self-study guide for anyone who wishes to get acquainted with geometry of Banach spaces of polynomials with applications. This text is specifically designed to be appealing and accessible to the reader, and provides a general overview on the topic together with new and interesting directions of research. The text also contains original results and material never published before.Features· Comprehensive review on the geometry of spaces of polynomials.· Visually attractive and accessible presentation, with over 75 explanatory figures.· Contains many examples illustrating the results and techniques appearing in the book.· Open (and deep!) questions within the area are provided so that the interested reader can begin doing independent research using the techniques presented in the text.· It also features original results by the authors.

This brief presents a global perspective on the geometry of spaces of polynomials. Its particular focus is on polynomial spaces of dimension 3, providing, in that case, a graphical representation of the unit ball. Also, the extreme points in the unit ball of several polynomial spaces are characterized. Finally, a number of applications to obtain sharp classical polynomial inequalities are presented.The study performed is the first ever complete account on the geometry of the unit ball of polynomial spaces. Nowadays there are hundreds of research papers on this topic and our work gathers the state of the art of the main and/or relevant results up to now. This book is intended for a broad audience, including undergraduate and graduate students, junior and senior researchers and it also serves as a source book for consultation. In addition to that, we made this work visually attractive by including in it over 50 original figures in order to help in the understanding of allthe results and techniques included in the book.

Over 100 years ago Harald Bohr identified a deep problem about the convergence of Dirichlet series, and introduced an ingenious idea relating Dirichlet series and holomorphic functions in high dimensions. Elaborating on this work, almost twnety years later Bohnenblust and Hille solved the problem posed by Bohr. In recent years there has been a substantial revival of interest in the research area opened up by these early contributions. This involves the intertwining of the classical work with modern functional analysis, harmonic analysis, infinite dimensional holomorphy and probability theory as well as analytic number theory. New challenging research problems have crystallized and been solved in recent decades. The goal of this book is to describe in detail some of the key elements of this new research area to a wide audience. The approach is based on three pillars: Dirichlet series, infinite dimensional holomorphy and harmonic analysis.