Guy David - Böcker

This book proposes new notions of coherent geometric structure. Fractal patterns have emerged in many contexts, but what exactly is a "pattern" and what is not? How can one make precise the structures lying within objects and the relationships between them? The foundations laid herein provide a fresh approach to a familiar field. From this emerges a wide range of open problems, large and small, and a variety of examples with diverse connections to other parts of mathematics.One of the main features of the present text is that the basic framework is completely new. This makes it easier for people to get into the field. There are many open problems, with plenty of opportunities that are likely to be close at hand, particularly as concerns the exploration of examples. On the other hand the general framework is quite broad and provides the possibility for future discoveries of some magnitude.Fractual geometries can arise in many different ways mathematically, but there is not so much general language for making comparisons. This book provides some tools for doing this, and a place where researchers in different areas can find common ground and basic information.

Wavelets are a recently developed tool for the analysis and synthesis of functions. This text begins with an introduction to the theory of wavelets and limits itself to the detailed construction of various orthonormal bases of wavelets. The second section of the book focuses on a criterion for the L2-boundedness of singular integral operators, known as the T(b) theorem. The book concludes with a survey of recent attempts to understand the geometry of subsets of Rn, in which analogues of the Cauchy kernel define bounded operators.