# Fourier and Harmonic Analysis

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This is a previous supplementary module, not currently scheduled to run this session.

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## Description

This course explores fundamentals of modern analysis developed in the early to mid twentieth century, centred around the study of the Fourier transform.

A major achievement of early twentieth century mathematics was to place Fourier analysis on a fiirm theoretical foundation. The definition of Fourier series/the Fourier transform, and the important Fourier inversion formulae, involve numerous limits (integrals, infinite sums, etc) and the attendant convergence questions tend to be rather subtle. The first part of the course deals with laying down the theoretical foundations through distribution theory and Lp spaces.

The Fourier series and transform provide indispensable tools for studying a wide variety of problems in analysis. The latter part of the course investigates the celebrated Calderon-Zygmund theory developed in the 1950s. This provides a remarkably powerful framework, leveraging the power of the Fourier transform to study various operators defined in terms of the frequencies of a given function. For instance, we investigate the classical Hilbert transform, which, roughly speaking, corresponds to a filters from acoustics: it takes an input function and removes all the parts which lie outside a certain bandwidth. We also introduce Littlewood-Paley theory, which provides another type of frequency decomposition of a function, this time into many different parts which each oscillate at a fixed scale. These ideas have had a fundamental impact on modern analysis, and particularly harmonic analysis and PDE.

Syllabus: The course is broken into three parts:

- The first part combines tools from measure theory and functional analysis to develop a robust theory of Fourier series and Fourier integrals. The focus is on defining these objects as bounded linear operators between Lp spaces. Topics include distribution theory, Plancherel's theorem, Fourier multipliers and the Hausdorff-Young inequality.
- The second part concerns certain geometric operators, such as the Hardy-Littlewood maximal function. The analysis involves the relatively simple geometry of Euclidean balls and cubes, which underpins various questions concerning the Fourier transform. Topics include real interpolation methods and the Hardy-Littlewood maximal theorem. These investigations lead to the important Calderon-Zygmund decomposition of an L1(Rn) function.
- The final part introduces the celebrated Calderon-Zygmund theory of the 1950s. This is used to prove the classical Riesz theorem on the Lp-boundedness of the Hilbert transform, to study Hormander multipliers and to prove the Littlewood-Paley theorem. To conclude, Littlewood-Paley theory is used to prove a (substantially!) weakened version of Carleson's celebrated theorem on almost everywhere convergence of Fourier series.