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MA433 Fourier Analysis

Lecturer: Professor Ian Melbourne

Term(s): Term 1

Status for Mathematics students: List C

Commitment: 30 lectures

Assessment: 3 hour exam

Prerequisites: Familiarity with measure theory at the level of MA359 Measure Theory. A knowledge of Hilbert spaces (e.g., MA3G7 Functional Analysis) is helpful but not necessary.

Leads To: Advanced courses in analysis and probability, for example MA4A2 Advanced Partial Differential Equations, MA4J0 Advanced Real Analysis, and MA911 Probability: Theory and Examples.

Content: Fourier analysis lies at the heart of many areas in mathematics. This course is about the applications of Fourier analytic methods to various problems in mathematics and sciences. The emphasis will be on developing the ability of using important tools and theorems to solve concrete problems, as well as getting a sense of doing formal calculations to predict/verify results. Topics will include:

  1. Fourier series of periodic functions, Gibbs phenomenon, Fejer and Dirichlet kernels, convergence properties, etc.
  2. Basic properties of the Fourier transform on R^d, including L^p theory.
  3. Topics on the Fourier inversion formula, including the Gauss-Weierstrass and Abel Poisson kernels, and connections to PDE.
  4. A selection of more advanced topics, including the Hilbert transform and an introduction to Singular Integrals.

Aims: The aim of the module is to convey an understanding of the basic techniques and results of Fourier analysis, and of their use in different areas of maths.

References (optional): The following books may also contain useful materials

- Stein, E. & Shakarchi, R. Fourier Analysis, an Introduction. Princeton University Press 2003.
- Duoandikoetxea, J. Fourier Analysis - American Mathematical Society 2001.
- Körner, T. Fourier Analysis, CUP 1988.
- Strichartz, R. A Guide to Distribution Theory and Fourier Transforms, CRC Press 1994.
- Folland, G. Real Analysis: Modern Techniques and their applications, Wiley 1999.
- Grafakos, L. Classical Fourier Analysis - Springer 2008.
- Grafakos, L. Modern Fourier Analysis - Springer 2008.
- Stein, E.M. Singular Integrals and differentiability properties of functions and differentiability properties of functions. Princeton Univesity Press.

Additional Resources

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