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This volume presents the recent theory of function spaces paying special attention to some developments in the last 10-15 years related to neighbouring areas such as numerics, signal processing, fractal analysis, etc. Local building blocks, in particular (non-smooth) atoms, quarks, wavelet bases and wavelet frames will be considered in detail and applied afterwards to diverse problems, including a local smoothness theory, spaces on Lipschitz domains, and fractal analysis.The book is essentially self-contained, although it might be also considered as the continuation of the two books of the author with the same title.