Project III

Fourier series and applications

Wilhelm Klingenberg

Description

This project will lay out the elementary properties of the the Fourier series and then move to an application such as the isoparametric problem or Jacobi's identity for the theta function, the heat flow, wave motion (e.g. Huygen's principle), Poisson's summation formula, Heisenberg's inequality, Minkowski's theorem in the geometry of numbers, Gauss law of quadratic reciprocity. The wikipedia article on Fourier series is an excellent source for an introduction to the subject, and there is considerable scope for the students to choose from applications in pure and applied mathematics. The classic treatise by Dym and McKean has many applications, and a more recent student-friendly source is the book by Pereyra and Ward.

Prerequisites

Analysis in Many Variables II , Complex Analysis II

Corequisite

Analysis III

Resources


H. Dym, H.P. McKean : Fourier Series and Integrals, Academic Press 1974
Enrique A. Gonzalez-Velasco (1992). "Connections in Mathematical Analysis: The Case of Fourier Series". American Mathematical Monthly 99 (5): 427 - 441
Evans, 1998: Partial differential equations
Elias M. Stein and Rami Shakarchi, Fourier analysis, Princeton lectures in analysis, 2003
Cristina Pereyra and Lesley A. Ward, Harmonic analysis: from Fourier to wavelets, AMS. Student mathematical library (2012)

Links

email: Wilhelm Klingenberg

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