# MA 522 Fourier Analysis and Applications (Spring 2013)

Lectures: Wed 9.30 -10.55 am, Fri 9.30 -10.55 am; Tutorial: Wed 6:35 -7.30 pm in Room 114

Informations about the course:

Content:

Basic Properties of Fourier Series:
Uniqueness of Fourier Series, Convolutions, Cesaro and Abel Summability, Fejer's
theorem, Poisson Kernel and Dirichlet problem in the
unit disc, Mean square Convergence, Example of Continuous functions with divergent Fourier series.

Distributions and Fourier Transforms: Calculus of Distributions, Schwartz class of
rapidly decreasing functions, Fourier transforms of rapidly decreasing functions,
Riemann
Lebesgue lemma, Fourier Inversion Theorem, Fourier transforms of Gaussians.
Tempered Distributions: Fourier transforms
of tempered distributions, Convolutions, Applications to PDEs (Laplace, Heat and
Wave Equations), Schrodinger-Equation and Uncertainty principle.

Paley-Wienner Theorems, Poisson Summation Formula: Radial Fourier transforms and
Bessel's functions. Hermite functions.

Optional Topics: Applications to PDEs, Wavelets and X-ray tomography. Applications to Number Theory.

Office Hour:
Tue 10.30-11.30 am

Text/References:

1. G. Bachman, L. Narici, E. Beckenstein, Fourier and Wavelet Analysis, Springer 2000.
2. R. Strichartz, A Guide to Distributions and Fourier Transforms, CRC Press.
3. E.M. Stein and R. Shakarchi, Fourier Analysis: An Introduction, Princeton
University Press, Princeton 2003.
4. I. Richards and H. Youn, Theory of Distributions and Non-technical Approach,
Cambridge University Press, Cambridge, 1990.
5. S. Kesavan, Topics in Functional Analysis and Applications, New Age Internatinal Pub., 1989.

## Assignments

1. From Ref. 3 page 36 example 2 and 3; exercises 2.2, 2.6, 2.11, 2.13 (c)

2. From Ref. 3 exercises 1.10, 1.11, 2.8, 3.8, 3.13

3. From Ref. 2 Chapter 1, exercises 2, 3, 4, 5, 6, 8, 11 link

4. From Ref. 2 Chapter 2, all exercises

5. From Ref. 2 Chapter 3, exercises 2, 3, 5, 6, 7, 8, 9, 10, 13, 14, 15

6. From Ref. 2 Chapter 3, exercises 1, 12, 16, 17, 18, 20, 21

7. From Ref. 2 Chapter 4, all exercises