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).
Reading Assignment: Chapter 7.
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)
Reading Assignment: Chapter 6.
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.
Return to: Department
of Mathematics and Statistics, University
of New Mexico
Last updated: January 14, 2015