## MATH 757: Fourier Analysis (Spring 2012)

Course Information:

Syllabus

Results on the Convergence of Fourier Series (Updated April 25)

Class presentations:
-Liz: 4.1 The Isoperimetric Inequality, on March 5.
-
Mark and Jon T.: 4.2, Weyl's Equidistribution, on March 7
-Brian: 5.2 Applications of the Fourier transform to PDEs, March 30
-Brandon: 4.3, A continuous function that is nowhere differentiable, April 11
-Erin: 5.3. Poisson's summation formula, April 16
-Melissa: Riesz-Thorin interpolation theorem, April 18

Homework 1 (Due February 22):
From the book: Chapter 1 Problem 1
Chapter 2 Exercises 5, 6, 8, 10, 11, 16, 17, 18, 19, 20, Problems 2, 3.
Not from the book:
hw1-757-spring12.pdf

Homework 2 (Due April 4)

From the book: Chapter 3 Exercises 8, 9, 10, 12, 13, 14, 15, 16, 17, 18
Chapter 5 Ex 3,4
Not from the book: hw2-757-spring12.pdf
(no need to do #2)
This homework is complete. Last problems added Feb 15 at 10:30pm.

Homework 3 (Due May 9)
From the book: Chapter 5 Exercises 9, 14, 20 (Shannon's Sampling Theorem), Problem 1.

This homework is complete (March 23 at 4:40pm)

Class diary:

PAST WEEKS:

Week of Jan 09. Chapter 1. Everybody should be familiar with Exercises: #1-3 by Friday. Exercises # 6-8 prove simple facts we used today in class.

Week of Jan 16. We finished Chapter 1 on Wednesday and started Chapter 2. On Friday we covered 2.2 (Uniqueness)

Week of Jan 23. Monday: Examples 1 and 4 from 2.1, with more detail than the book. Wednesday: Example 5 in Section 2.1, all of Section 2.3 (Convolutions), and started Exercise 9. Friday: Finished Ex 9 and started Section 2.4 (Good Kernels)

Week of Jan 30. Monday: Proved Theorem 4.1, finished Section 2.4, defined Cesaro and Abel summability. Wednesday: Section 2.5, Cesaro and Abel summability of Fourier series.  Friday: Complete solution to the Dirichlet problem for the Laplace equation on the disc (including pointwise convergence, which is not in the book).

Week of Feb 6. Monday: Finished two lemmas from Friday and started Chapter 3. Wednesday: Lē theory of Fourier Series. Thursday and Friday: Pointwise convergence of Fourier series (Dini's Criterium, rate of convergence, jump discontinuities and Gibbs phenomenon)

Week of Feb 13. Monday: Ch.3, Sec 2.2: A continuous function with divergent Fourier series at 0. Wednesday: Started Ch.5, the Fourier transform (sections 1.1, 1.2). Friday: Ch 5. Section 1.3 and Proposition 1.2.

Week of Feb 20. Wednesday: Ch. 5, Secs 1.4 and 1.5. Friday: Extension of the Fourier transform to Lē and to Lp with 1<p<2 (not in the book).

Week of Feb 27. Monday: Riemann-Lebesgue, Applications of the Fourier transform to PDE, start Theory of Distributions (not in book). Friday: Comments about HW 1.

Week of March 5: Monday: The Isoperimetric Problem (4.1, presented by Liz). Wednesday: Weyl's Equidistribution (4.2, presented by Jon T. and Mark)

Week of March 12: Spring Break

Week of March 19: Theory of distributions (not in book); Partial sums of the Fourier transform: Norm convergence and pointwise convergence (not in book).

Week of March 26: Finished Partial sums of the Fourier transform (not in book). Applications to PDE (Brian)

Week of April 2: Chapter 6 (Fourier transform in R^n) and application to the wave equation.