Intended for students with a beginning knowledge of mathematical analysis, this first volume, in a three-part introduction to Fourier analysis, introduces the core areas of mathematical analysis while also illustrating the organic unity between them. It includes numerous examples and applications.
Microlocal Analysis And Complex Fourier Analysis:
This is the first monograph that discusses in detail the interactions between Fourier analysis and dynamic economic theories, in particular, business cycles. Many economic theories have analyzed cyclical behaviors of economic variables. In this book, the focus is on a couple of trials: (1) the Kaldor theory and (2) the Slutsky effect. The Kaldor theory tries to explain business fluctuations in terms of nonlinear, 2nd-order ordinary differential equations (ODEs). In order to explain periodic behaviors of a solution, the Hopf-bifurcation theorem frequently plays a key role. Slutsky´s idea is to look at the periodic movement as an overlapping effect of random shocks. The Slutsky process is a weakly stationary process, the periodic (or almost periodic) behavior of which can be analyzed by the Bochner theorem. The goal of this book is to give a comprehensive and rigorous justification of these ideas. Therefore, the aim is first to give a complete theory that supports the Hopf theorem and to prove the existence of periodic solutions of ODEs; and second to explain the mathematical structure of the Bochner theorem and its relation to periodic (or almost periodic) behaviors of weakly stationary processes. Although these two targets are the principal ones, a large number of results from Fourier analysis must be prepared in order to reach these goals. The basic concepts and results from classical as well as generalized Fourier analysis are provided in a systematic way. Prospective readers are assumed to have sufficient knowledge of real, complex analysis. However, necessary economic concepts are explained in the text, making this book accessible even to readers without a background in economics.
Das Buch ist eine um 156 Seiten erweiterte Neufassung des vom Autor 1996 bei Vieweg erschienenen Titels ´´Fourieranalysis, Distributionen und Anwendungen´´, der seit langem vergriffen ist. Rezensionen zu diesem Titel waren:´´...Besonders wertvoll sind die zahlreichen Beispiele, die sowohl klassische als auch moderne Anwendungen (z.B. in der Signalanalysis) beschreiben. Das Buch ist also vor allem für Mathematiker(innen) geeignet, die ein wichtiges Werkzeug der anwendungsorientierten Analysis kennenlernen wollen.´´H.G.Feichtinger, Wien (Monatshefte für Mathematik 126, 1998)´´...Zusammenfassend empfehle ich das Buch als eine hervorragende Einführung in die klassische und distributionelle harmonische Analysis und in die Distributionentheorie.´´N.Ortner, Innsbruck (Intern. Math. Nachrichten 181, 1999)
In recent years, the Fourier analysis methods have expereinced a growing interest in the study of partial differential equations. In particular, those techniques based on the Littlewood-Paley decomposition have proved to be very efficient for the study of evolution equations. The present book aims at presenting self-contained, state- of- the- art models of those techniques with applications to different classes of partial differential equations: transport, heat, wave and Schrödinger equations. It also offers more sophisticated models originating from fluid mechanics (in particular the incompressible and compressible Navier-Stokes equations) or general relativity. It is either directed to anyone with a good undergraduate level of knowledge in analysis or useful for experts who are eager to know the benefit that one might gain from Fourier analysis when dealing with nonlinear partial differential equations.
Fourier Analysis and Convexity:
Fourier Analysis and Approximation: