FOURIER ANALYSIS AND APPLICATIONS
Summer 2008

Instructor: Nguyen Cong Phuc
Purdue University, USA
Email: pcnguyen@math.purdue.edu

Time: From July 1, 2008 to July 21, 2008. Class meetings:

Lectures: MW&F from 1:00pm to 3:00pm.

Problem sections: Tu&Th from 1:00pm to 3:30pm. These will be conducted by a TA. Language: English.

I. Course description: This is an elementary introduction to Fourier analysis exposing basic facts about Fourier series and Fourier transforms with their applications to partial differential equations and number theory.

II. Prerequisites: Analysis 1. No knowledge of Lebesgue integration theory is needed as we carry out our treatment of Fourier series and transforms in the context of Riemann integrable functions.

III. Textbook: Fourier Analysis An Introduction by Elias M. Stein and Rami Shakarchi, Princeton University Press, Princeton and Oxford, 2003.

VI. Contents:

Chapter 1. Basic Properties of Fourier Series: Definition, uniqueness, Fourier series and convolutions, Ces`aro and Abel summability.

Chapter 2. Convergence of Fourier Series: Mean-square convergence, pointwise convergence.

Chapter 3. Some Applications of Fourier Series: Isoperimetric inequality, Weyl’s equidistribution theorem, heat equations on the circle.

Chapter 4. The Fourier Transform on R: Elementary theory, applications to partial differential equations, Poisson summation formula, Heisenberg uncertainty principle.

Chapter 5. The Fourier Transform on R^d: Elementary theory, applications to partial differential equations.