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 author's book "Fractals and Spectra" (MMA 91). It is directed to
mathematicians and (theoretical) physicists interested in the topics
indicated and, in particular, how they are interrelated.