### MATH 472/572 - HOMEWORK PROBLEMS - Fall 2012

The homework refers to exercises from the textbook.

Homework 1 (due 8/28/12) :
Chapter 1 - 1.2, 1.4, 1.7.

Homework 2 (due 9/4/12) :
Work in teams, with Yuridia-David a new team, all the others the same).
Turn in problems 1. and [2. or 3.]:

1. Show that uniform limit of continuous functions is continuous (Theorem 2.59).
2. Show that if f is continuous on [a,b] then f is Riemann integrable on [a,b].
3. Show that if f_n is a sequence of Riemann integrable functions on [a,b] that converges uniformly to f on [a,b] then f is Riemann Integrable and the limit of the integrals is the integral of f (Thm 2.53).

Homework 3 (due 9/11/12) :
Chapter 3 Exercise 3.11 in p.62 and exercise 4.18 p.89.

Homework 4 (due 9/18/12) :
This homework is a group work (due Tu 9/18/2012): 5 teams (3 people on each team):

1. Team 1: David, and Cameron
2. Team 2: Matt, Audrey.
3. Team 3: Dusty, Jared, and James
4. Team 4: Wang, Anastassiya, and Christopher
5. Team 5: Vishnu, Yuri, and Craig.
One solution set turn in per team, all the members must be aware of what is being submitted, please acknowledge sources used: books, internet, etc.
1. Exercise 3.25 (plucked string)
2. Exercise 4.5 (the L^1 norms of D_N are not uniformly (in N) bounded, they grow like log N.

Homework 5 (due on Thursday 9/27/2012):

1. Exercises 4.16 and 4.17.

Homework 6 (due Tuesday October 9, 2012):

1. Proof of the Cauchy-Schwarz inequality
2. Exercise 5.8 (show that the space of little ell 2 of square summable sequences is complete).

Homework 7 (due 10/23/12):

1. Exercises: 6.3, 6.13, 6.14, 6.19, 6.35.

Homework 8 (due 11/06/12):

1. Exercise 7.11 (Gaussian is its own Fourier transform)
2. Exercise 7.20 (Time Frequency dictionary convolution becomes product and viceversa)
3. Exercise 7.29 (practice use of Time Frequency dictionary)

Homework 9 (due 11/13/2012):

1. Exercise 8.26 (finish Time-Frequency dictionary in S')
2. Exercise 8.30 (calculus with the delta distribution)
3. Exercise 8.33 (approximations of identity converge to delta distribution)
4. Bonus: Exercise 8.46 (assuming Theorem 8.44)

Final Projects Fest

1. Tuesday Dec 4th, 2012 (with snacks and coffee)
1. 9:00-9:30 Monsters I: Nowhere Differentiable Functions (Yuri and Audrey)
2. 9:35-10:05 Monsters II: A continuous functions whose partial Fourier sums diverge at a zero (David and Matt)
3. 10:10-10:40 Weyl's equidistribution theorem (Dusty)
2. Thursday Dec 6th, 2012 (with snacks and coffee)
1. 9:00-9:25 Fourier Analysis on Groups (Cameron)
2. 9:30-9:55 Averaging and Summability Methods (Bishnu and Craig)
3. 9:55-10:10 Snacks and fill evaluation forms
4. 10:10-10:40 Isoperimetric Problem (James and Chris)
3. Tuesday Dec 11th, 2012 (with snacks and coffee)
1. 8:30-8:45 Bowl of kernels (Wang)
2. 8:50-9:30 Wavelets in action (Anastassiya and Wang)
3. The end ;-)