### MATH 472/572 - HOMEWORK PROBLEMS - Fall 2013

Final Projects

Tuesday Nov 26, 2013

• 7:30-8:00 -- Eli and Cullen Gibb's Phenomenon slides, report

• 8:05-8:35 -- Mahshid Multipole Methods and Sampling project

• 8:40-9:10 -- Denise Spectral methods for solving PDEs slides, report

Tuesday Dec 3, 2013

• 7:30-8:00 Nihil -- Weyl's equidistribution theorem report

• 8:05-8:35 Nuriye, Cesar and Patrick -- Kernels in R, slides

• 8:40-9:10 Cullen and Michael -- Wavelets decomposition of EECs project

The homework refers to exercises from the textbook.

Homework 10 ***LAST*** (due Thursday Nov 14, 2013)
Exercise 9.22 (show Shannon function is a wavelet),
Find the Shannon Multiresolution Analysis (it can be described completely like the Haar multiresolution analysis): Exercises: 10.6 (see Example 10.5) and 10.39 (see Example 10.38).
Reading assignment: Chapter 9: Sections 9.3, and Chapter 10: Section 10.1, 10.2 and 10.4.

Homework 9 (due Thursday Oct 31, 2013)
Exercise 8.26 (Time frequency dictionary for tempered distributions),
Exercise 8.33 (approximations of the identity converge to the delta distribution).

Homework 8 (due Tuesday October 22, 2013):
Work individually or in groups as you please:
Exercises 7.11, 7.13, 7.19 and 7.20 (use the inversion formula (Theorem 7.28 in Section 7.6 to show (j) )

Homework 7 (due Thursday 10/10/2013):
Work individually or in groups, do only one of the following (I would like some of you to choose the first, some of you to choose the second, you are welcome to try both but it is not needed):

• Exercise 6.29 (scrambling miracle)
• Exercises 6.30 and 6.32

Homework 6 (due Tuesday 10/01/2013):
Work individually or in groups, do only one of the following:

• 1) Exercise 5.13 (prove Pythagorean Theorem, Cauchy-Schwarz and the triangle inequeality),
• 2) Exercise 5.8 (space of square summable sequences is a complete vector space),
• 3) Prove Theorem 5.31 in a different order than proposed in the book (as discussed today in class).

Homework 5 (due Tuesday 9/24/2013):
You are welcome to work in groups of two, same as last time, or you may switch around:
Exercise 4.16 [note there are two typos in the Hint: k > 0 (or k >= 1), and k=1 (instead of k=0)], Exercises 4.37 and for the computer inclined 4.40 for the Poisson kernel.
Reading assignment: Sections 4.2 and 4.5.

Homework 4 (due Thursday 9/12/2013):
Exercises 4.5 (you can provide numerical evidence by reproducing Figure 4.2), and 4.30 (closed formula for the Fejer Kernel).
This is group work, how about the following teams?

• 1) Cullen and Denise,
• 2) Mahshid and Nikhil,
• 3) Nuriye and Patrick,
• 4) Eli and Britton,
• 5) Cesar and Michael. If some of the teams want to work think together is fine, but write separately your solutions, and make sure both team members understand most of the work if not all.

Homework 3 (due Thursday after Labor Day 9/5/2013):
Exercise 3.18, and 3.25. Do either 3.19 or prove that uniform convergence preserves continuity (Theorem 2.59).