Fourier Analysis and Distributions : A First Course with Applications / by Rolf Brigola.

Format:

Book

Author/Creator:

Brigola, Rolf.

Series:

Texts in Applied Mathematics, 2196-9949 ; 79

Language:

English

Subjects (All):

Fourier analysis.

Mathematical analysis.

Fourier Analysis.

Analysis.

Local Subjects:

Fourier Analysis.

Analysis.

Physical Description:

1 online resource (1033 pages)

Edition:

1st ed. 2025.

Place of Publication:

Cham : Springer Nature Switzerland : Imprint: Springer, 2025.

Summary:

This comprehensive book offers an accessible introduction to Fourier analysis and distribution theory, blending classical mathematical theory with a wide range of practical applications. Designed for undergraduate and beginning Master's students in mathematics and engineering. Key Features: Balanced Approach: The book is structured to include both theoretical and application-based chapters, providing readers with a solid understanding of the fundamentals alongside real-world scenarios. Diverse Applications: Topics include Fourier series, ordinary differential equations, AC circuit calculations, heat and wave equations, digital signal processing, and image compression. These applications demonstrate the versatility of Fourier analysis in solving complex problems in engineering, physics, and computational sciences. Advanced Topics: The text covers convolution theorems, linear filters, the Shannon Sampling Theorem, multi-carrier transmission with OFDM, wavelets, and a first insight into quantum mechanics. It also introduces readers to the finite element method (FEM) and offers an elementary proof of the Malgrange-Ehrenpreis theorem, showcasing advanced concepts in a clear and approachable manner. Practical Insights: Includes a detailed discussion of Hilbert spaces, orthonormal systems, and their applications to topics like the periodic table in chemistry and the structure of water molecules. The book also explores continuous and discrete wavelet transforms, providing insights into modern data compression and denoising techniques. Comprehensive Support: Appendices cover essential theorems in function theory and Lebesgue integration, complete with solutions to exercises, a reference list, and an index. With its focus on practical applications, clear explanations, and a wealth of examples, Fourier Analysis and Distributions bridges the gap between classical theory and modern computational methods. This text will appeal to students and practitioners looking to deepen their understanding of Fourier analysis and its far-reaching implications in science and engineering.

Contents:

Preface

Introduction

Trigonometric Polynomials, Fourier Coefficients

Fourier Series

Calculating with Fourier Series

Application Examples for Fourier Series

Discrete Fourier Transforms, First Applications

Convergence of Fourier Series

Fundamentals of Distribution Theory

Application Examples for Distributions

The Fourier Transform

Basics of Linear Filters

Further Applications of the Fourier Transform

The Malgrange-Ehrenpreis Theorem

Outlook on Further Concepts

A The Residue Theorem and the Fundamental Theorem of Algebra

B Tools from Integration Theory

C Solutions to the Exercises

References

List of Symbols and Physical Quantities

Index.

ISBN:

9783031813115

3031813111