Course Information:
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.
Not from the book: hw3-757-spring12.pdf (click here for tex file)
This homework is
complete (March 23 at 4:40pm)
Class
diary:
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.