Harmonic Analysis

Department: 
MATH
Course Number: 
7337
Hours - Lecture: 
3
Hours - Lab: 
0
Hours - Recitation: 
0
Hours - Total Credit: 
3
Typical Scheduling: 
Every fall semester

Fourier analysis on the torus and Euclidean space.

Prerequisites: 

MATH 6337 and MATH 6338

Course Text: 

At the level of Katznelson, “An Introduction to Harmonic Analysis” or Muscalu and Schlag, “Classical and Multilinear Harmonic Analysis.”

Topic Outline: 
  • Fourier series.
  • L 1 and L 2 theory.
  • Approximate identities, completeness of exponentials, Weyl Equidistribution.
  • Convergence of Fourier series.
  • Duality between smoothness and decay.
  • L p theory (Hausdorff–Young Theorem).
  • The Fourier transform and its applications.
  • Additional topics at instructor’s discretion as time permits. Typical additional topics may include the following (and others):
    • Paley–Wiener Theorems.
    • Sobolev spaces.
    • Poincare inequalities and spherical harmonics.
    • Uncertainty principles.
    • Fourier transform of distributions.
    • Fourier transform of measures, Bochner’s Theorem, Wiener’s Theorem.
    • Wiener’s algebra and Wiener’s Lemma.
    • Ideal structure. T
    • ime-frequency analysis (local Fourier analysis).
    • Littlewood–Paley theory, wavelets.