Gallier J., Quaintance J. Aspects of Harmonic Analysis and Representation Theory

University of Pennsylvania (Dept. of Computer and Information Science), 2019. — 764 p.The question that motivated writing this book is:What is the Fourier transform?We were quite surprised by how involved the answer is, and how much mathematics is needed to answer it, from measure theory, integration theory, some functional analysis, to some representation theory.
The main topic of this book is the Fourier transform and Fourier series, both understood in a broad sense. Historically, trigonometric series were first used to solve equations arising in physics, such as the wave equation or the heat equation. D’Alembert used trigonometric series (1747) to solve the equation of a vibrating string, elaborated by Euler a year later, and then solved in a different way essentially using Fourier series by D. Bernoulli (1753). However it was Fourier who introduced and developed Fourier series in order to solve the heat equation, in a sequence of works on heat diffusion, starting in 1807, and culminating with his famous book, Th´eorie analytique de la chaleur, published in 1822.Function Spaces Often Encountered.
The Riemann Integral.
Measure Theory; Basic Notions.
Integration.
Radon Measures on Locally Compact Spaces.
The Haar Measure and Convolution.
The Fourier Transform and Cotransform on Tn , Zn , Rn.
Normed Algebras and Spectral Theory.
Analysis on Locally Compact Abelian Groups.
Hilbert Algebras.
Representations of Locally Compact Groups.
Analysis on Compact Groups and Representations.
Induced Representations.
Harmonic Analysis on Gelfand Pairs.
A Topology.
B Vector Norms and Matrix Norms.
C Basics of Groups and Group Actions.
D Hilbert Spaces.