An Invitation to Fourier Analysis and Distribution Theory
Buch in deiner Nähe kaufen
Beschreibung
This book provides a rigorous introduction to Fourier analysis and the theory of distributions without presupposing an advanced background in functional analysis or measure-theoretic integration.
The guiding principle throughout is to present the material in an elementary and direct manner without sacrificing mathematical precision or depth. After introducing Fourier series and their fundamental properties in the first chapter, the main ideas of Fourier analysis are developed first for non-periodic functions on Euclidean space, where many concepts can be presented more transparently, and are subsequently transferred to the periodic setting. Likewise, distribution theory is initially developed in the framework of tempered distributions, which allows for a simpler exposition and is particularly well suited to applications in partial differential equations, including those discussed in Chapter 7. Distributions on open subsets of Euclidean space are then introduced via localization, emphasizing their inherently local character. The final chapter presents an elementary proof of the Schwartz kernel theorem based on expansions in Hermite functions, from which the tensor product of distributions is obtained as an immediate consequence. Each chapter concludes with a collection of exercises ranging from routine applications to more challenging problems.
This book provides a rigorous introduction to Fourier analysis and the theory of distributions without presupposing an advanced background in functional analysis or measure-theoretic integration.
The guiding principle throughout is to present the material in an elementary and direct manner without sacrificing mathematical precision or depth. After introducing Fourier series and their fundamental properties in the first chapter, the main ideas of Fourier analysis are developed first for non-periodic functions on Euclidean space, where many concepts can be presented more transparently, and are subsequently transferred to the periodic setting. Likewise, distribution theory is initially developed in the framework of tempered distributions, which allows for a simpler exposition and is particularly well suited to applications in partial differential equations, including those discussed in Chapter 7. Distributions on open subsets of Euclidean space are then introduced via localization, emphasizing their inherently local character. The final chapter presents an elementary proof of the Schwartz kernel theorem based on expansions in Hermite functions, from which the tensor product of distributions is obtained as an immediate consequence. Each chapter concludes with a collection of exercises ranging from routine applications to more challenging problems.
Covers both Fourier analysis and distribution theory Accessible to students with only basic real and functional analysis as background Takes an original approach to distribution theory not found in other books
Autor*in
Detlef Müller
Themen in »An Invitation to Fourier Analysis and Distribution Theory«
Fourier transform Fourier series Plancherel's theorem Spectral decomposition Poisson summation formula Tempered distributions Convolution Distributions with compact support Seminorms Schwarz functions Hermite functions Schwarz space Applications of distribution theory to PDE Heat equation Laplace operator