Gallier J., Quaintance J. Aspects of Harmonic Analysis on Locally Compact Abelian Groups

World Scientific, 2024. — 760 p.The Fourier transform is a "tool" used in engineering and computer vision to model periodic phenomena. Starting with the basics of measure theory and integration, this book delves into the harmonic analysis of locally compact abelian groups. It provides an in-depth tour of the beautiful theory of the Fourier transform based on the results of Gelfand, Pontrjagin, and Andre Weil in a manner accessible to an undergraduate student who has taken linear algebra and introductory real analysis.
Highlights of this book include the Bochner integral, the Haar measure, Radon functionals, the theory of Fourier analysis on the circle, and the theory of the discrete Fourier transform. After studying this book, the reader will have the preparation necessary for understanding the Peter–Weyl theorems for complete, separable Hilbert algebras, a key theoretical concept used in the construction of Gelfand pairs and equivariant convolutional neural networks.Preface
Introduction
Function Spaces Often Encountered
The Riemann Integral
Measure Theory; Basic Notions
Integration
The Fourier Transform and the Fourier Cotransform on 𝕋n, ℤn, ℝn
Radon Functionals and Radon Measures on Locally Compact Spaces
The Haar Measure and Convolution
Normed Algebras and Spectral Theory
Harmonic Analysis on Locally Compact Abelian Groups
Appendices:
Topology
Vector Norms and Matrix Norms
Basics of Groups and Group Actions
Hilbert Spaces
Well-Ordered Sets, Ordinals, Cardinals, Alephs
Bibliography
Symbol Index
Index