Fourier Analysis and Distributions A First Course with Applications

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.

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.

Rolf Brigola studied and received his doctorate in mathematics at the University of Erlangen-Nürnberg, Germany. and was research assistent in the faculty of mathematics at the University of Regensburg, Germany, in the fields of functional analysis and stochastic processes. He has more than 10 years work experience in development and application of mathematical methods (electrodynamics, control engineering, elasticity problems) in the energy technology division of Siemens AG in Erlangen and in development of operating system and application software for digital communication systems in the communications technology division of Siemens AG in Munich, Germany. Since 1988 he has worked as professor for mathematics with students of applied mathematics, physics and engineering disciplines. He is the author of the previous textbook "Fourieranalysis, Distributionen und Anwendungen" in German with Vieweg in 1996.