The Structure of Functions This book deals with the constructive Weierstrassian approach to the theory of function spaces and various applications. The first chapter is devoted to a detailed study of quarkonial (subatomic) decompositions of functions and distributions on euclidean spaces domains manifolds and fractals. This approach combines the advantages of atomic and wavelet representations. It paves the way to sharp inequalities and embeddings in function spaces spectral theory of fractal elliptic operators and a regularity theory of some semi linear equations. The book is self contained although some parts may be considered as a continuation of the authors book Fractals and Spectra. It is directed to mathematicians and (theoretical) physicists interested in the topics indicated and in particular how they are interrelated. The book under review can be regarded as a continuation of [his book on Fractals and spectra 1997] (..) There are many sections named: comments preparations motivations discussions and so on. These parts of the book seem to be very interesting and valuable. They help the reader to deal with the main course. (Mathematical Reviews)