Requirements/prerequisites: prerequisite: MA2223 and MA2224
Duration: Micaelmas term, 11 weeks

Number of lectures per week: 3 lectures including tutorials per week

Assessment:

ECTS credits: 5
End-of-year Examination: A 2-hour examination in Trinity term,

Description: Harmonic Analysis is one of the most successful and beautiful areas of mathematics. From its origins in Fourier series, it has expanded in various ways - singular integral operators, complex analysis, group representation theory, operator theory.

Topics:

Fourier Series: Origins. Convergence of Cesaro means. Mean-square convergence.

Pointwise convergence (for smooth classes). Failure of pointwise convergence.

Weyl's equidistribution theorem.

Fourier Transform: Definition, inversion, Plancherel formula.

Textbooks:

• Fourier Analysis, An introduction, by E. M. Stein and R. Shakarchi, Princeton University Press.

• An introduction to harmonic analysis, by Y. Katznelson, Dover.

• Theory of discrete and continuous Fourier analysis by H. J. Weaver (includes the theory of Lebesgue integration)

Learning Outcomes: On successful completion of this module, students will be able to:

• compute the Fourier coefficients of a given function;

• compare two functions by comparing their Fourier coefficients;

• recognize and apply good kernels;

• apply Cesaro and Abel summability to Fourier series;

• apply different convergence methods to Fourier series;

• apply Fourier series to some problems in analysis;

• compute the Fourier transform of a (suitable) function, its inverse, understand and apply the Plancherel formula.