Fourier Analysis — FAN

Instructor: Dr. Arpad TOTH

Text:  Stein-Shakarchi: Fourier Analysis (Princeton University Press, 2003)

Course Description: Fourier series and the Fourier transform originated from physical applications, but turned into a major motivating force in the transformation of Analysis from the "calculus of analytic functions" into the study of much more general functions such as Lebesgue measurable functions, distributions, the concept of the integral, of length, and their properties. Fourier analysis has a long and rich history, but is still a very active area of research.

Topics will cover roughly the first 5 chapters of the book by Stein and Shakarchi but some of the more advanced topics may be skipped. In particular, we will go over

• the origins and essential properties of Fourier series, including various convergence results;
• the most important analytical tool, the so called "convolution";
• Fourier's inversion formula;
• several applications. These include partial differential equations but are much broader in scope.
• Some highlights of theorems covered are the isoperimetric inequality, the Weierstrass approximation theorem, and Heisenberg's uncertainty principle.

Students who take this class will understand what Fourier series and the Fourier transform are, what analytic techniques arise from their use and how they can be applied in a variety of situations. It will be especially useful for those students who want a deeper understanding of analysis, in particular, want to see the historical motivation for the study of Lebesgue measure and integral, or who are interested in applications and desire to know more than mere formulas for Fourier coefficients.

Prerequisite: The course requires a solid understanding of convergence of sequences and series. If the formal definitions of convergence of sequences and convergence of series make you hesitant, you should probably take ANT and not this course.
You also need to be familiar with the Riemann integral, in the way usually covered in an Honors Calculus or Introductory Analysis course.
Familiarity with complex numbers is needed as well.