Harmonic Analysis: A Gentle Introduction

Material is presented in a unified way, avoiding the "cookbook" character of many texts.  For example, in chapter one an interesting boundary value problem is solved.  The discussion then leads naturally to harmonic functions, periodic functions and Fourier series.  The various ways such series can converge are investigated in three separate chapters.

Students are brought into the discussion by providing them with the results to their exercises.  Solutions to starred exercises throughout the text are provided in an appendix.

The structure and format of the text is flexible, allowing the instructor to tailor it to the needs of the course and the needs of the students.

0.   Preliminaries
      0.1   Set Theory
      0.2   Relations and Functions
      0.3   The Real Number Theory
      0.4   The Complex Number System
      0.5   Analysis

1.   Classical Harmonic Analysis
      1.1   The Dirichlet Problem for a Disk
      1.2   Continuous Funtions on the Unit Circle
      1.3   The Method of Fourier
      1.4   Uniform Convergence
      1.5   The Formulas of Euler
      1.6   Cesaro Convergence
      1.7   Fejer's Theorem
      1.8   At Last the Solution

2.   Extensions of the Classical Theory
      2.1   Functions on (-p, p)
      2.2   Functions on Other Intervals
      2.3   Functions with Special Properties
      2.4   Pointwise Convergence of Fourier Series

3.   Fourier Series in Hilbert Space
      3.1   Normed Vector Spaces
      3.2   Convergence in Normed Spaces
      3.3   Inner Product Spaces
      3.4   Infinite Orthonormal Sets, Hilbert Spaces
      3.5   The Completion (Appendix B)
      3.6   Wavelets

4.   The Fourier Transform
      4.1   The Fourier Transform on Z
      4.2   Invertible Elements in ℓ¹ (Z)
      4.3   The Fourier Transform of R
      4.4    Naive Group Theory
      4.5    Not so Naive Group Theory
      4.6    Finite Fourier Transforms
      4.7    An Application
      4.8    Some Algebraic Matters
      4.9    Prime Numbers
      4.10  Euler's Phi Function

5.   Abstract Algebra
     5.1   Groups
     5.2   Morphisms
     5.3   Rings
     5.4   Fields

Appendix A:  Linear Algebra
Appendix B:  The Completion
Appendix C:  Solution to starred problems 

Harmonic Analysis: A Gentle Introduction is primarily intended for mathematics majors in their junior or senior year.  It is also appropriate for both under-graduate and graduate students in various engineering and physical science departments. Prerequisites include advanced calculus and linear algebra.