Harmonic Analysis (Math 360)

Spring 2011

Meetings: Tuesday, Thursday 11:00-12:15

Instructor: Akram Aldroubi

Prerequisites: Real Analysis (Math 330 a-b)

References: 

Christopher Heil, Introduction to Harmonic Analysis (upcoming)

Mark Pinsky, Introduction to Fourier Analysis and Wavelets,

Walter Rudin, Fourier Analysis on Groups,

E. Stein, Singular integrals and differentiability properties of functions,

E. Stein, Harmonic analysis, Real Variable methods, Orthogonality and

Oscillatory Integrals,

L. Grafakos, Classical and modern Fourier analysis.

Syllabus: 

This course will focus on the mathematical theory of modern harmonic analysis. It will cover the basic theory of Fourier and harmonic analysis: Fourier series; Fourier integral; Fourier analysis on locally compact Abelian groups; Multiresolution approximations; Wavelet Theory. The focus will be on main results and mastering proof techniques necessary for research.  

Content: 

Chapter 1: Fourier Series

1. Motivation;
2. Fourier coefficients; Fourier series; 
3. Convergence of the Fourier series;
4. Properties of the Fourier series.
5. Wiener's Lemma on absolutely convergent Fourier series.

Chapter 2: Fourier Transform

1. The Fourier integral of L^1 functions;
2. Inversion and properties of the Fourier transforms;
3. Plancherel Theorem;
4. The Fourier transform on L^p spaces, and distributions;

Chapter 3: Fourier Analysis on Groups

1. Haar Measure on a LCA group G;
2. The dual group  and the Fourier integral;
3. The Inversion Theorem;

Chapter 4: Fourier Analysis

1. Hausdorff-Young Inequality;
2. The Hilbert transform;
3. Hardy-Littlewood Maximal Functions;
4. Marcinkiewicz Interpolation Theorem;
5. Singular integrals and the Calderon-Zygmund decomposition.

Chapter 5: Wavelet Transform

1. Multiresolution Approximations of L²;
2. Wavelet bases and frames of L²;
3. Haar wavelet and Shannon wavelet;
4. Compactly supported wavelets and Spline wavelets.