## Mathematics 541 (Harmonic Analysis I)

### Fall 2010

**Instructor: I. Laba (Math Bldg 200, 604-822-4457,
ilaba(at)math.ubc.ca).
**

Lectures: MWF 2-3, MATX 1102.

Office hours: M 12-1, F 1-2, and by appointment.

This course will provide an introduction to harmonic analysis on Euclidean spaces, with
applications to number theory, PDE and geometric measure theory.

A list of topics we covered, with references to sources, is available here.

**Homework assignments:**
- Homework 1, due on
**Friday, September 24**
- Homework 2, due on
**Friday, October 8**
- In #1(a), the interval [0,1] should be replaced by [-1/2,1/2].
- In #2(b), you may assume that f is in L
^{1}(**T**), so that the
Fourier coefficients are well defined. (This is implicit in Katznelson's book.)
- A typo in #3: replace n by x in the integrand in the hint.

- Homework 3, due on
**Friday, October 29**
- Homework 4, due on
**Friday, November 19**
- Homework 5, due on
**Friday, December 3**

**Tentative topics:**
- Fourier series
- Summability (Cesaro, Abel) and convergence
- Applications to number theory (Weyl's equidistribution theorem),

- The Fourier transform on
**R**^{n}
- The Schwartz space
- Fourier inversion and the Plancherel formula
- L
^{p} spaces and the Hausdorff-Young formula
- The Poisson summation formula
- The uncertainty principle
- Applications to PDE theory

- Fourier transforms of singular measures
- The stationary phase method
- Fourier transforms of sphere-carried measures
- Application: counting lattice points in large discs

- Introduction to the Hilbert transform
- The Riesz and Kolmogorov theorems
- L
^{p} convergence of the Fourier transform

**Recommended textbooks:**
*Fourier Analysis*, J. Duoandikoetxea, American Mathematical Society, 2001
*An Introduction to Harmonic Analysis*, Y. Katznelson, Cambridge University
Press, 2004
*Harmonic Analysis: Real-Variable Methods, Orthogonality, and
Oscillatory Integrals*, E.M. Stein, Princeton Univ. Press, 1993
*Fourier Analysis: An Introduction*, E.M. Stein and R. Shakarchi,
Princeton University Press, 2003
*Lectures on Harmonic Analysis*, T. Wolff, American Mathematical Society, 2003

**Prerequisites:** Basic real and functional analysis, including measure theory
and integration, Banach spaces, linear operators, L^{p} spaces.
Math 420/507 and 421/510 cover all the required background.

**Your course grade** will be based on 5 problem sets, tentatively due on September 24,
October 8, October 27, November 17 and December 1. To allow for minor absences and short illnesses,
the lowest score will be dropped; each of the remaining 4
problem sets will be worth 25% of your grade. There will be no final exam.

**Math 542**, offered in Spring 2011, will cover several current research areas
in harmonic analysis and geometric measure theory. Tentatively, this will include Kakeya
sets, restriction theory, maximal estimates and differentiation theorems for submanifolds
of **R**^{n}, Hausdorff dimension and Fourier dimension (if this does not get
covered in Math 541), Salem sets, projection theorems and distance sets.