MATH 472/572 - HOMEWORK PROBLEMS - Sring 2015

Numbering refers to our textbook.

Homework 9 (due on Tuesday 11/22/16 ):

• Chapter 9: Exercise 9.22 (The Shannon wavelet).
• Chapter 10: Exercise 10.6 (The Shannon MRA).
• Chapter 10: Exercise 10.39 (do only Shannon low and high-pass filters).

Homework 8 (due on Tuesday Nov 1st):

• Chapter 8: Exercise 8.6 (Find Fourier transform of Fejer and Poisson kernels).
• Chapter 8: Exercise 8.26 (Time-frequency dictionary in S').
• Chapter 8: Exercise 8.39 (Poisson summation formula applied to Fejer and Poisson kernels).

Homework 7 (due on Thursday October 20, 2016):

• Chapter 7: Exercise 7.29 (use Time-Frequency dictionary to verify identity we used in class).
• Chapter 7: Project 7.8 (c)(d) (Fejer, Poisson, Gaussian, and Heat kernels are approximations of the identity, Dirichlet kernels are not but appear naturally when analysing partial Fourier integrals).

Homework 6 (due on Tuesday Oct 11, 2016):

• Chapter 7: Exercise 7.11 (Gaussian is a Schwartz function, and is its own Fourier transform).
• Chapter 7: Exercise 7.19 (Schwartz class is closed under convolution).
• Chapter 7: Prove Theorem 7.24 (Convolution of a Schwartz function f with an approximation of the identity converges uniformly to f).

Homework 5 (due on Thursday Sep 29, 2016):
[the or is a mathematical or, you can do one or the other or both!)

• Chapter 6: Exercises 6.30 AND 6.32 (interplay of circular convolution, discrete Fourier transform DFT, and shift invariance).
• Chapter 6: Exercise 6.37 (compute DFT of Haar vectors and "verify" that the more localized in space the least localized in frequency and vice versa), OR do Exercise 6.44 (rewriting the Haar vectors as dilation and shifts of the Haar function h_1 in our original labeling h_n, with 0<= n <= N-1.)
• Chapter 6: Exercise 6.35 ({v_n} is an o.n. basis in C^N iff {\hat{V_n}} is an o.n. basis in C^N, \hat{v_n} is the DFT of v_n, a vector in C^N).

Homework 4 (due Thursday Sep 22, 2016):

• Chapter 6: Exercise 6.3 (verify that the Fourier vectors in C^N form and orthonormal basis).
• Chapter 6: Exercise 6.13 ("orthonormality" property of dual basis).

Homework 3 (due on Thursday Sep 15, 2016):

• Chapter 5: Exercise 5.8 (show that little ell-2 (Z) is a complete space, A^N is the Nth sequence in a sequence of sequences, not to be confused with a power).
• Chapter 5: Exercise 5.17 (inner product in L^2 coincides with inner product in little ell-2 of the Fourier coefficients).

Homework 2 (due on Thursday 9/8/16) :

• Chapter 4: Exercise 4.5 (Show that analitically that L^1 norm of D_N grows like log N or provide numerical evidence),
• Chapter 4: Exercise 4.16 (Convolution improves smoothness).
Reading assignment: Chapter 4 (go back to Chapter 2 as needed)

Homework 1 (due on Thursday 9/1/16) :

• Chapter 3: Exercise 3.11 (averages of sequences),
• Chapter 3: 3.25 (plucked string) or show Weierstrass M-Test (Theorem 2.55).
Reading assignment: Chapter 3 (go back to Chapter 2 as needed)

Warmup Homework (due on Thursday 8/25/16) :

• Chapter 1: Exercises 1.2,
• Chapter 1: Exercise 1.7 and 1.11,
• Chapter 1: Exercise 1.13.
Reading assignment: Chapters 1 and 2.