Fourier Analysis

Fourier Analysis (also known as Harmonic Analysis) is the study of functions by decomposing them with respect to nice bases. Classical Fourier Analysis decomposes periodic functions into linear combinations of sines and cosines. While this decomposition is useful in the field of Analysis (for example, in solving certain partial differential equations), Fourier Analysis has also proved effective in other fields, including: signal processing/time series data analysis, geometry and topology, and number theory.

Our first goal in this project will be to build a solid foundation in the theory of Fourier Analysis by studying the Fourier transform including the following topics:

• Definition of the Fourier transform.
• Properties of the Fourier transform.
• Convolution and the Fourier transform.
• Differentiation and the Fourier transform.
• Fourier inversion.

After laying these foundations together, you would be free to choose your preferred direction in which to continue your investigation. Some possible directions include:

• Using the Fourier transform to solve partial differential equations.
• Exploring other transforms (e.g. Hilbert, Radon, Riesz transforms) and their connections to the Fourier transform.
• Using the Fourier transform to study finite groups, and/or connect these ideas to signal processing.
• Exploring uses of the Fourier transform in number theory, including Dirichlet’s theorem: if \(q\) and \(l\) are relatively prime positive integers, then there are infinitely many primes of the form \(l + kq\) with \(k\) an integer.

Essential Prior Modules : Complex Analysis II.

Essential Companion Modules : Analysis III.

Depending on your preferred direction of the project, the module Partial Differential Equations III could be useful but is not essential.

• L.C. Evans, Partial Differential Equations, AMS, 2010.
• Y. Katznelson, An Introduction to Harmonic Analysis, Cambridge University Press, 2012.
• Richard S. Laugesen , Harmonic Analysis.
• M. C. Pereyra, L. Ward, Harmonic Analysis: from Fourier to Wavelets, AMS 2012.
• E. M. Stein, R. Shakarchi, Fourier Analysis: An Introduction, Princeton University Press, 2003.