1Chapter-1 Fourier Analysis
317.1 Introduction
21.1 Introduction
327.2 Bessel Functions and Fourier-Bessel Series
31.2 Historical Overview
337.3 Legendre Polynomials and Fourier-Legendre Series
41.3 Basic Concepts and Motivation
347.4 Applications in Quantum Mechanics and Electromagnetic Theory
5Chapter-2 Fourier Series
35Chapter-8 Nonlinear Fourier Analysis and Applications
62.1 Introduction
368.1 Introduction
72.2 Representation of Periodic Functions
378.2 Nonlinear Fourier Transforms
82.3 Convergence and Gibbs Phenomenon
388.3 Soliton Solutions and Integrable Systems
92.4 Fourier Coefficients and Parseval’s Identity
398.4 Applications in Nonlinear Optics and Wave Propagation
10Chapter-3 Orthogonal Functions and Fourier Transform
40Chapter-9 Generalised Fourier Series and Transforms
113.1 Introduction
419.1 Introduction
123.2 Orthogonality and Inner Product Spaces
429.2 Fourier Series for Non-Periodic Functions
133.3 Fourier Transform and Inverse Transform
439.3 Generalised Fourier Transforms (e.g., Mellin Transform, Hankel Transform)
143.4 Properties of Fourier Transforms
449.4 Applications in Signal Processing and Spectral Analysis
15Chapter-4 Fourier Analysis on the Real Line
45Chapter-10 Discrete-Time Fourier Analysis
164.1 Introduction
4610.1 Introduction
174.2 Fourier Integrals and Inversion Formula
4710.2 Discrete-Time Fourier Series
184.3 Convolution and Plancherel Theorem
48Chapter-11 Sampling Theory and the Nyquist-Shannon Sampling Theorem
194.4 Applications in Signal Processing and Filtering
4911.1 Introduction
20Chapter-5 Discrete Fourier Transform (DFT) and Fast Fourier Transform (FFT)
5011.2 Nyquist Criteria for Sampling Rates
215.1 Introduction
5111.3 Aliasing and Reconstruction of Signals
225.2 Definition and Properties of DFT
5211.4 Practical Implications in Digital Signal Processing
235.3 FFT Algorithm and Efficiency
53Chapter-12 Harmonic Analysis on Locally Compact Abelian Groups
245.4 Applications in Digital Signal Processing and Data Compression
5412.1 Introduction
25Chapter-6 Fourier Analysis in Higher Dimensions
5512.2 Pontryagin Duality and Fourier Analysis on Compact Groups
266.1 Introduction
5612.3 Fourier Analysis on Locally Compact Abelian Groups
276.2 Fourier Series and Transforms in Multiple Variables
5712.4 Applications in Probability Theory and Quantum Mechanics
286.3 Spherical Harmonics and Laplace’s Equation
58Glossary
296.4 Applications in Partial Differential Equations and Heat Conduction
59Index
30Chapter-7 Special Functions and Applications
