Fourier Analysis on Finite Abelian Groups

• Main Author: Luong, Bao. (Author)
• Corporate Author: SpringerLink (Online service)
• Format: Electronic
• Language: English
• Published: Boston : Birkhũser Boston, 2009.
• Series: Applied and Numerical Harmonic Analysis
• Subjects: Mathematics.; Group theory.; Fourier analysis.; Fourier Analysis.; Group Theory and Generalizations.
• Online Access: https://ezaccess.library.uitm.edu.my/login?url=http://dx.doi.org/10.1007/978-0-8176-4916-6

Fourier analysis has been the inspiration for a technological wave of advances in fields such as imaging processing, financial modeling, cryptography, algorithms, and sequence design. This self-contained book provides a thorough look at the Fourier transform, one of the most useful tools in applied...

Table of Contents:

• Preface
• Overview
• Chapter 1: Foundation Material
• Results from Group Theory
• Quadratic Congruences
• Chebyshev Systems of Functions
• Chapter 2: The Fourier Transform
• A Special Class of Linear Operators
• Characters
• The Orthogonal Relations for Characters
• The Fourier Transform
• The Fourier Transform of Periodic Functions
• The Inverse Fourier Transform
• The Inversion Formula
• Matrices of the Fourier Transform
• Iterated Fourier Transform
• Is the Fourier Transform a Self-Adjoint Operator?
• The Convolutions Operator
• Banach Algebra
• The Uncertainty Principle
• The Tensor Decomposition
• The Tensor Decomposition of Vector Spaces
• The Fourier Transform and Isometries
• Reduction to Finite Cyclic Groups
• Symmetric and Antisymmetric Functions
• Eigenvalues and Eigenvectors
• Spectrak Theorem
• Ergodic Theorem
• Multiplicities of Eigenvalues
• The Quantum Fourier Transform
• Chapter 3: Quadratic Sums
• 1. The Number G_n(1)
• Reduction Formulas.