MATH 472/572 - HOMEWORK PROBLEMS - Fall 2019

Numbering refers to our textbook.

Homework 10 (due on Thursday 11/7/19): Do at least two of the problems.

• Chapter 10: Exercise 10.5 (the Shannon MRA).
• Chapter 10: Exercise 10.39 (Shannon low-pass and high-pass filters satisfy a QMF condition).
• Chapter 10: Exercise 10.8 (integer translates phi_{j,k} for fixed j, k in Z, form an orthonormal basis for the approximation spaces V_j of an orthogonal MRA).
• Chapter 10: Exercise 10.17 (detail subspace W_j is a dilation of W_0).

Homework 9 (due on Thursday 10/31/19): Do at least two of the three problems.

• Chapter 9: Exercise 9.8 (Fourier transform of a Gabor function is a Gabor function).
• Chapter 9: Exercise 9.14 (the continuous Gabor transform).
• Chapter 9: Exercise 9.22 (Shannon functions form an orthonormal basis).

Homework 8 (due on Friday 10/25/19):

• Chapter 8: Exercise 8.26 (time-frequency dictionary in S').
• Chapter 8: Exercise 8.30 (derivatives and antiderivatives of the delta distribution).

Homework 7 (due on Thursday 10/17/19): Do at least three of the following five problems.

• Chapter 7: Exercise 7.11 (Gaussian is a Schwartz function, and is its own Fourier transform).
• Chapter 7: Exercise 7.13(b)-(g) (Time frequency dictionary in the Schwartz class).
• Chapter 7: Exercises 7.19 and 7.20 (Shwartz class closed under convolution and interplay between convolution and product. Note that you'll need the inversion formula for (j).)
• Chapter 7: Prove Theorem 7.24 (Convolution of a Schwartz function f with an approximation of the identity converges uniformly to f).
• Chapter 7: Exercise 7.32 (Polarization identity in S(R)).

Homework 6 (due on Tuesday 10/8/19): Do at least three of the following four itemized assignments.

• Chapter 6: Exercises 6.30 AND 6.32 (interplay of circular convolution with discrete Fourier transform DFT and shift invariance).
• 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).
• 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).
• Chapter 6: Exercise 6.44 (rewrite the Haar vectors as dilation and shifts of the Haar function h_1 in our original labeling h_n, with 0<= n <= N-1.)

Homework 5 (due Thursday 9/26/19): Choose at least three problems from the following list.

• Chapter 5: Exercise 5.32 (equivalent conditions for a complete orthonormal system).
• Chapter 5: Exercise 5.34 (orthogonal projection and best approximation lemma in L^2(T)).
• Chapter 6: Exercise 6.3 (verify that the Fourier vectors in C^N form and orthonormal basis).
• Chapter 6: Exercises 6.13 and 6.14 (Orthonormality property and uniqueness of dual basis

Homework 4 (due on Thursday 9/19/19): Choose at least three problems from the following list.

• Prove the Cauchy-Schwarz inequality for any inner-product vector space. Use it to deduce the triangle inequality for the norm induced by the inner product.
• Chapter 2: Exercise 2.27 (L^p(I) spaces, I bounded interval are nested).
• Chapter 5: Exercise 5.8 (show that little ell-2 (Z) is a complete inner-product vector 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).

Reading Assignment: Chapter 5, and you may find useful information in Sections 2.1.2-2.1.3 and in Appendixes A.1 and A.2.

Homework 3 (due on Thursday 9/12/19): Choose at least two problems from the following list of three..

• Chapter 4: Exercise 4.16 (Convolution improves smoothness).
• Chapter 4: Exercises 4.17 and 4.18 (Convolution with trig polynomial returns trig polynomial, translations correspond to modulations).
• Chapter 4: Exercises 4.30 and 4.31 (closed formula for the Fejer kernel and they are good kernels).

Homework 2 (due on Thursday 9/5/19): Choose at least two problems from the following list.

• Chapter 3: Exercise 3.22 (decay of Fourier coefficients of Lipschitz functions).
• Chapter 3: Exercise 3.25 (about the plucked string).
• Chapter 4: Exercise 4.5 (Show analitically that L^1 norm of D_N grows like log N or provide numerical evidence).
Reading assignment: Chapter 3 and 4 (go back to Chapter 2 as needed).

Homework 1 (due on Thursday 8/29/19): Choose at least two problems from the following list.

• Chapter 2: Prove the Weierstarss M-Test (Theorem 2.55).
• Chapter 3: Exercise 3.1 (Matlab exploration of Gibb's phenomenon).
• Chapter 3: Exercise 3.11 (sequences and averages).