MATH 472/572 - HOMEWORK PROBLEMS - Fall 2021

Numbering refers to our textbook.

Homework 10 (due on Tuesday 11/16/21 at 11:59pm) Last!!!!: Do at least two of the problems. You can do the additional problem for bonus points.

  • Chapter 10: Exercise 10.6 (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 with scaling function phi).
  • Chapter 10: Exercise 10.17 (detail subspace W_j is an appropriate dilation of W_0).

    Reading Assignment: Chapters 10-11.

    Homework 9 (due on Thursday 11/04/21 at 11:59pm): Do at least two of the three problems. You can do the additional problem for bonus points.

  • 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 function is a wavelet).

    Reading Assignment: Chapter 9.

    Homework 8 (due on Thursday 10/28/21 at 11:59pm): Do at least three of the following six problems. You can do the additional problems for bonus points.

  • Chapter 8: Exercise 8.6 (Fourier transform of Fejer and Poisson kernels).
  • Chapter 8: Exercise 8.11 (Fourier inversion formula and Plancherel hold when both f and f^ are continuous functions of moderate decrease).
  • Chapter 8: Exercise 8.18 (Polynomials induce tempered distributions).
  • Chapter 8: Exercise 8.26 (time-frequency dictionary in S').
  • Chapter 8: Exercise 8.30 (derivatives and antiderivatives of the delta distribution).

    Reading Assignment: Chapter 8.

    Homework 7 (due on Thursday 10/21/21 at 11:59pm): Do at least three of the following six problems. You can do the additional problems for bonus points.

  • Chapter 7: Exercise 7.7 (Bump functions are in the Schwartz class).
  • 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 (Schwartz 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 7.27 (Multiplication formula in C^N).

    Reading Assignment: Chapter 7.

    Homework 6 (due on Thursday 10/7/21 at 11:59pm): Choose at least two problems from the following list. You can do the additional problem for bonus points.

  • 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).
  • Chapter 6: Exercise 6.48 (Fast Haar tansform is of order N).

    Reading Assignment: Chapter 6, Sections 6.5-6.7.

    Homework 5 (due Thursday 9/30/21 at 11:59pm): Choose at least three (sorry it said two but not in the email nor in Learn) problems from the following list. You can do the additional problem for bonus points.

  • 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).
  • Chapter 6: Exercise 6.29 (a miracle with the scrambling matrices in the FFT decomposition).
  • Chapter 6: Exercises 6.30 AND 6.32 (interplay of circular convolution with discrete Fourier transform DFT and shift invariance).

    Reading Assignment: Sections 6.1-6.4

    Homework 4 (due on Thursday 9/23/21 at 11:59pm): Choose at least two problems from the following list. You can do the additional problem for bonus points.

  • 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).
  • Chapter 5: Exercise 5.34 (orthogonal projection and best approximation lemma in L^2(T)).

    Reading Assignment: Chapter 5 and Appendixes A.1 and A.2. In particular Sections 5.4 and A.2.3 and A.2.4 should be quite useful.

    Homework 3 (due on Thursday 9/16/21 at 11:59pm): Choose at least two problems from the following list. You can do additional problems for bonus points.

  • 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 on Fourier side to modulations).
  • Chapter 5: Exercise 5.13 (norm induced by an inner product is a norm. Prove the Pythagorean Theorem and the Cauchy-Schwarz Inequality).
  • Chapter 5: Exercise 5.8 (show that little ell-2 (Z) is a complete inner-product vector space (where A ^N={a_{n,N}: n in Z} is for each N>0 the Nth sequence in a Cauchy sequence of sequences, not to be confused with a power).

    Reading Assignment: Chapter 4 and Chapter 5. You may find useful information in Sections 2.1.2-2.1.3 and in Appendixes A.1 and A.2.

    Homework 2 (due on Thursday 9/9/21 at 11:59pm): Choose at least two problems from the following list. You can do additional problems for bonus points.

  • Chapter 3: Exercise 3.19 - appropriate rate of decay of Fourier coefficients implies certain smoothness (First step in the induction to prove Theorem 3.16, once done, complete the induction. The base case is presented in the book. Note that in 3.16 and Corollary 3.17 one should assume that f is continuous).
  • Chapter 3: Exercise 3.25 (about the plucked string in Example 3.24 and the rates of convergence of its partial Fourier sum).
  • Chapter 4: Exercise 4.5 (Show analytically that L^1 norm of D_N grows like log N or provide numerical evidence).
  • Chapter 4: Exercises 4.30 and 4.31 (closed formula for the Fejer kernel and they are good kernels).

    Reading assignment: Chapter 3 and 4 (go back to Chapter 2 as needed).

    Homework 1 (due in Learn on Thursday 9/2/21 at 11:59pm): Do at least three problems. You can do additional problems for bonus points.

  • Chapter 1: Exercise 1.4 (show that Cauchy's function is infinitely many times differentiable).
  • Chapter 1: Select one of the models in 1.5 Project: Other Physical Models and find solutions following the approach in Section 1.4.
  • Chapter 2: Exercise 2.29 (finite and countable sets have measure zero) and 2.31 (Cantor Set).
  • Chapter 3: Exercise 3.1 (Matlab exploration of Gibb's phenomenon). For additional insight, look at Section 3.4 Project: Gibbs Phenomenon.
  • Chapter 3: Exercise 3.11 (sequences and averages). Adapt your argument to the setting of uniform convergence: if a sequence of continuous complex-valued functions {f_n: n>0} defined on a closed *and bounded* interval I of the real numbers converges uniformly to f on I, then the sequence of arithmetic averages g_n=(f_1 + f_2 +...+ f_n)/n converges uniformly to f on I. [Note a posteriori try to come up with a counterexample if the interval is not assumed to be bounded or closed.]

    Reading assignment: Chapter 3.

    Warmup Homework (due on Thursday 8/26/21):

  • Chapter 1: Exercise 1.2. A convergent trigonometric series can be written in terms of sine and cosines (with coefficients b_n and c_n as in equation (1.1)) or in terms of complex exponentials (with coefficients a_n as in equation (1.2)). Write formulas relating coefficients {a_n: n in Z} to coefficients {b_n, c_n: n in N} and vice versa.
  • Chapter 1: verify the orthonormality of the complex exponentials (that is make sure you believe equation (1.6) specifically the last equality needs some justification).

    Reading assignment: Chapter 1.

    Return to: Department of Mathematics and Statistics, University of New Mexico

    Last updated: September 23, 2021