### MATH 472/572 - HOMEWORK PROBLEMS - Fall 2017

Numbering refers to our textbook.

Final Projects

• The uncertainty principle on finite Abelian groups by Sulgi Bak (12/12/17)
• A study of Gibbs Phenomenon by Douglas Brunson (12/12/17)
• Wavelets and their applications to music by Ian Canavan (12/7/17)
• System identification using finite impulse response filters by Victor Nevarez (12/5/17)
• Harmonic analysis and partial differential equations by Ari Rappaport (12/5/17)
• Fourier analysis on finite Abelian groups by Zack Stevens (11/30/17)

Homework 9 (due on November 21, 2017):

• 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 8 (due on Nov 9, 2017):

• Chapter 9: Exercise 9.22 (Shannon functions form an orthonormal basis).
• Chapter 10: Exercise 10.5 (the Shannon MRA).
• Chapter 10: Exercise 10.39 (Shannon low-pass and high-pass filters satisfy a QMF condition).

Homework 7 (due on October 26, 2017):

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

Homework 6 (due on Thursday Oct 5, 2017): Do all three problems.

• 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 28, 2017): Do at least two of the following three itemized assignments [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 21, 2017):

• Chapter 6: Exercise 6.3 (verify that the Fourier vectors in C^N form and orthonormal basis).

Homework 3 (due on Thursday 9/14 2017): Choose at least two 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 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).
• Chapter 5: Exercise 5.32 (equivalent conditions for a complete orthonormal system).

Homework 2 (due on Thursday 9/7/17): Choose at least two problems from the following list.

• 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).
• Chapter 4: Exercise 4.30 (closed formula for the Fejer kernel).
Reading assignment: Chapter 4 (go back to Chapter 2 as needed).

Homework 1 (due on Thursday 8/31/17): 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).
• Chapter 3: Exercise 3.18 (decay of Fourier coefficients implies smoothness).