Lecturer: Professor Ian Melbourne
Term(s): Term 1
Commitment: 30 lectures
Assessment: Oral exam
Assumed knowledge: Familiarity with measure theory at the level of MA359 Measure Theory especially Fubini's Theorem, Dominated and Monotone Convergence Theorems.
Useful background: Further knowledge of Functional Analysis such as: MA3G7 Functional Analysis I and MA3G8 Functional Analysis II is helpful but not necessary. Topics such as norms of bounded linear operators will be reviewed in the module. Some basics about Hilbert spaces will also be reviewed in the module. The uniform boundedness principle will be stated without proof, but the other major results from functional analysis are not used.
Synergies: The following modules go well together with Fourier Analysis:
Leads to: The following modules have this module listed as assumed knowledge or useful background:
- MA627 Ergodic Theory
- MA6M2 Mathematics of Inverse Problems
- MA6A7 Quantum Mechanics: Basic Principles and Probabilistic Methods
- MA6J0 Advanced Real Analysis
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:
- Fourier series of periodic functions, Gibbs phenomenon, Fejer and Dirichlet kernels, convergence properties, etc
- Basic properties of the Fourier transform on R^d, including L^p theory
- Topics on the Fourier inversion formula, including the Gauss-Weierstrass and Abel Poisson kernels, and connections to PDE
- 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 University Press.