ASPECTS HARMONIC ANALYSIS ON LOCALLY COMPACT ABELIAN GROUPS

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.

Contents:
PrefaceIntroductionFunction Spaces Often EncounteredThe Riemann IntegralMeasure Theory; Basic NotionsIntegrationThe Fourier Transform and the Fourier Cotransform on 𝕋n, ℤn, ℝnRadon Functionals and Radon Measures on Locally Compact SpacesThe Haar Measure and ConvolutionNormed Algebras and Spectral TheoryHarmonic Analysis on Locally Compact Abelian GroupsAppendices:TopologyVector Norms and Matrix NormsBasics of Groups and Group ActionsHilbert SpacesWell-Ordered Sets, Ordinals, Cardinals, AlephsBibliographySymbol IndexIndex
Readership: (1) Senior year undergraduate math major, or first to second year graduate masters/PhD student of mathematics, engineering or computer vision who is interested learning about harmonic analysis. (2) Appropriate for mathematical courses on classical and functional analysis, medical imaging and measurement, advanced applied mathematics, mathematical analysis, representation of continuous groups, and mathematics for Engineering. (3) Supplementary reading for courses in signal/image process, deep learning/equivariant convolution neural networks, heat conduction, automatic control, acoustic, optics, and structural analysis. (4) Reference/self study manual for the engineer or computer scientist who needs to apply the Fourier in their research.

Key Features: Gentle but rigorous introduction to the mathematical underpinning of Fourier transform and harmonic analysis: With illustrations and annotations to help the reader understand the material Proof sketches and references are given for certain proofs to avoid overwhelming the reader Self-contained with prerequisite materials in the appendices; suitable as a textbook or for self-study Extensive bibliography, suitable also as a reference book Provide sufficient background to understand the classic but more advanced texts Prepare sufficient background to understand the cutting-edge theory in equivariant convolutional neural network results