MATM18 Fourier Analysis is an optional course for a Master of Science degree in mathematics. It is an introductory course on Fourier Analysis.
Fourier series were first used by the French Mathematician Joseph Fourier around 1804 in his study of the heat flow. They are one of the most powerful mathematical concepts for applications, ubiquitous in Physics, Electrical Engineering, and many branches of Computing. However, also modern-day Pure Mathematics, for example Number Theory, Geometry and Representation Theory, is permeated by the study of the Fourier transform and its generalizations. This course will introduce some of the tools of Fourier Analysis and present some applications.
Our main concern will be the question as to in which sense a function may be represented by its Fourier series or transform. In addition, we will look at applications in Differential Equations, Number Theory, Real and Complex Analysis.
We will concentrate on the following topics:
How can a function be recovered (pointwise) from its Fourier series ? (convolution kernels, Dirichlet kernel, Fejer kernel, Poisson kernel, Fatou's Theorem ) Some applications of Fourier Series in Analysis and Number theory (the isoperimetric inequality, Weyl's equidistribution theorem, a continuous function which is nowhere differentiable ) Fourier transform and the Paley-Wiener Theorem Finite Fourier transform
Comments on Prerequisites : The course "Analytic Functions" is a prerequisite. However, only a relatively small part of the material in this course will be required, so if you have already taken a Complex Analysis course (for example at your home university, in case you are an Erasmus student), you can still attend this course. The courses "Topology" and "Integration" are useful, but not mandatory.
The teaching consists of lectures and seminars. Compulsory hand-in exercises may be given.
The course is assessed through a written and an oral examination.
E.M.Stein, R.Shakarchi, Fourier analysis, Princeton Lectures in Analysis I, Princeton University Press, 2003.
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