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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.
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.