Fractals and Spectra: Related to Fourier Analysis and Function Spaces [Paperback]

This book deals with the symbiotic relationship between the theory of function spaces, fractal geometry, and spectral theory of (fractal) pseudodifferential operators as it has emerged quite recently. Most of the presented material is published here for the first time.

This book deals with the symbiotic relationship between the theory of function spaces, fractal geometry, and spectral theory of (fractal) pseudodifferential operators. It investigates distributions of eigenvalues of fractal (pseudo)differential operators.n This book deals with several aspects of fractal geometry in ? which are closely connected with Fourier analysis, function spaces, and appropriate (pseudo)differ- tial operators. It emerged quite recently that some modern techniques in the theory of function spaces are intimately related to methods in fractal geometry. Special attention is paid to spectral properties of fractal (pseudo)differential operators; in particular we shall play the drum with a fractal layer. In some sense this book may be considered as the fractal twin of [ET96], where we developed adequate methods to handle spectral problems of degenerate n pseudodifferential operators in ? and in bounded domains. Besides a few special properties of function spaces we relied there on sharp estimates of entropy numbers of compact embeddings between these spaces and their relations to the distribution of eigenvalues. Some of the main assertions of the present book are based on just these techniques but now in a fractal setting. Since virtually nothing of these new methods is available in literature, a substantial part of what we have to say deals with recent developments in the theory of function spaces, also for their own sake. In this respect the book might also be considered as a continuation of [Tri83] and [Tri92].Fractals.- ?p-spaces.- Function spaces on ?n.- Function spaces on and of fractals.- Spectra of fractal pseudodifferential operators.Hans Triebel ist Professor l£ª