This book may be considered as the continuation of the monographs [Tri?]and [Tri?] with the same title. It deals with the theory of function spaces of type s s B and F as it stands at the beginning of this century. These two scales of pq pq spacescovermanywell-knownspacesoffunctionsanddistributionssuchasH� older- Zygmundspaces,(fractionalandclassical)Sobolevspaces,BesovspacesandHardy spaces. On the one hand this book is essentially self-contained. On the other hand we concentrate principally on those developments in recent times which are related to the nowadays numerous applications of function spaces to some neighboring areas such as numerics, signal processing and fractal analysis, to mention only a few of them. Chapter 1 in [Tri?] is a self-contained historically-oriented survey of the function spaces considered and their roots up to the beginning of the 1990s entitled How to measure smoothness. Chapter 1 of the present book has the same heading indicating continuity. As far as the history is concerned we will now be very brief, restricting ourselves to the essentials needed to make this book self-contained and readable. We complement [Tri?], Chapter 1, by a few points omitted there. But otherwise we jump to the 1990s, describing more recent developments. Some of them will be treated later on in detail.

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This book deals with the recent theory of function spaces as it stands now. Special attention is paid to some developments in the last 10–15 years which are closely related to the nowadays numerous applications of the theory of function spaces to some neighbouring areas such as numerics, signal processing and fractal analysis. In particular, typical building blocks as (non-smooth) atoms, quarks, wavelet bases and wavelet frames are discussed in detail and applied afterwards to some outstanding problems of the recent theory of function spaces such as a local smoothness theory, fractal measures, fractal analysis, spaces on Lipschitz domains and on quasi-metric spaces.

The book is essentially self-contained, although it might also be considered as the continuation of the two previous books of the author with the same title which appeared as volumes 78 and 84 in this book series. It is directed to mathematicians working in analysis, numerics and fractal geometry, and to (theoretical) physicists interested in related subjects such as signal processing.

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