An invitation to Fourier analysis and distribution theory / Detlef Müller

Format:

Book

Author/Creator:

Müller, Detlef, 1954- author.

Series:

Universitext 2191-6675

Universitext, 2191-6675

Language:

English

Subjects (All):

Fourier analysis.

Theory of distributions (Functional analysis).

Fourier Analysis.

Medical Subjects:

Fourier Analysis.

Physical Description:

1 online resource

Place of Publication:

Cham, Switzerland : Springer, [2026]

Summary:

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

Contents:

The basic idea of Fourier analysis : expansion of periodic functions into trigonometric series

Fourier transform and convolution on ℝn

Fourier series and the Poisson summation formula

Tempered distributions

Distributions in open subsets of ℝn

Distributions with compact support

Fundamental solutions

On the regularity theory of linear partial differential equations : the singular support and hypoellipticity

The Schwartz kernel theorem, and the tensor product of distributions

Notes:

Includes bibliographical references and index

Online resource; title from PDF title page (Springer Nature Link, viewed September 2, 2026)

ISBN:

9783032315502

3032315506

OCLC:

1610816199

Access Restriction:

Restricted for use by site license