### MATH 472/572 - HOMEWORK PROBLEMS - Sring 2015

Homework 5 (due Tue March 17, 2015) You can work individually or in groups of 2-3. Make sure you include all names of team mates and you include all references used (books, internet sites, friends, etc)

• Exercise 7.11 (the Gaussian is its own Fourier transform).
• Exercise 7.13 (time frequency dictionary for S(R)).
• Exercise 7.19 (S(R) is closed under convolution. Assume known Property (i) in the time frequency dictionary).
• Exercise 7.30 (justify interchange of limit and integral without using the Lebesgue Differentiation Theorem).

Homework 4 (due Tu Feb 24) You can work individually or in groups of 2-3. Make sure you include all names of team mates and you include all references used (books, internet sites, friends, etc)

• 1) Exercise 6.3 (show that the trigonometric vectors are orthonormal)
• 2) Show that orthonormality implies linear independence.
• 3)Exercise 6.30 (Fourier coefficients of circular convolution is product of Fourier coefficients, and use it to get Fast convolution of vectors)
• 4) Exercise 6.37 (Discrete Fourier transform of Haar vectors, exploring localization both in space and frequency)

Homework 3 (due Th Feb 5, 2015) You can work individually or in groups of 2-3. Make sure you include all names of team mates and you include all references used (books, internet sites, friends, etc)

• Exercise 4.5 (check that the L^1 norms of the Dirichlet kernels grow logarithmically)
• Exercise 4.18 (Time frequency Dictionary: translation vs modulation)
• Exercise 4.22 (good kernels dilating a good function- check typo in errata page posted on class webpage)
• Exercise 4.30 (closed formula for the Fejer kernels)
• Chapter 5: show that if f,g are in L^2(T) then their inner product is equal to the inner product in ell-2(Z) of the sequences of their Fourier coefficients.
Reading Assignment: Chapter 4 and 5.

Homework 2 (due on Tuesday 1/27/15) :

• Chapter 3: Exercises 3.11 (averages of sequences) and 3.27 (plucked string)
Reading assignment: Chapter 3 (go back to Chapter 2 as needed)

Homework 1 (due on Tuesday 1/20/15) :

• Chapter 1: Exercises 1.2 and 1.4.
• Calculate the Fourier coefficients for f, the function defined as f(x)=1 for all 0<=x<=1, and f(x)=0 for 1
• Chapter 3: Exercise 3.1 (optional but if you know Matlab I encourage you to do it).
Reading assignment: Chapters 1 and 2.