MATH 472/572
- HOMEWORK PROBLEMS - Spring 2024
Numbering refers to our textbook.
Homework 11 (due on Tuesday 04/23/24 at 11:59pm) Last!!!!:
Chapter 10: Exercise 10.6 (the Shannon MRA, see Example 10.5).
Chapter 10: Exercise 10.39 (Shannon low-pass and high-pass filters satisfy a QMF condition).
Reading Assignment: Chapters 10-11.
Homework 10 (due on Friday 04/12/24 at 11:59pm): Do at least two of the problems. You can do the additional problem for bonus points.
Chapter 9: Exercise 9.26 (Dyadic intervals are nested or disjoint).
Chapter 9: Exercise 9.34 (integral average over parent interval is arithmetic average of integral averages over the dyadic children).
Chapter 9: Exercise 9.40 (Lebesgue Differentiation Theorem for continuous functions).
Reading Assignment: Chapter 9.
Homework 9 (due on Friday 04/05/24 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 Friday 03/29/24 at 11:59pm): Do at least three of the following five problems, including the time-frequency dictionary in S'. You can do the additional problems for bonus points.
Chapter 8: Exercise 8.4 and Exercise 8.6 (Fourier transform of Fejer and Poisson kernels. Note that both Fourier transforms are continuous functions of modeate decrease, if useful use Exercise 8.11).
Chapter 8: Exercise 8.11 (Fourier inversion formula and Plancherel when f and f^ are continuous functions of moderate decrease, you can use previous exercises like Exercises 8.7, 8.8, and 8.10).
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 Friday 3/22/24 at 11:59pm): Do at least three of the following list. You can do the additional problems for bonus points.
Chapter 7: Exercise 7.3 (the Schwartz class is closed under products, differentiation, multiplication by trigonometric functions and by polynomials (which are not Schwartz functions as they don't decay at infinity)).
Chapter 7: Exercise 7.11 (the 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(i) (Schwartz class closed under convolution and interplay between convolution and product).
Chapter 7: Prove Theorem 7.24 (Convolution of a Schwartz function f with an approximation of the identity converges uniformly to f).
Reading Assignment: Chapter 7.
Homework 6 (due on Friday 3/08/24 at 11:59pm): Choose at least two problems from the following list. You can do additional problems for bonus points.
Chapter 6: Exercises 6.20 and 6.35 (DFT preserves inner products, and o.n. basis are preserved under DFT).
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 6.42 (Haar vectors are orthonormal).
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 transform is of order N).
Reading Assignment: Chapter 6, Sections 6.5-6.7.
Homework 5 (due Friday 3/01/24 at 11:59pm):
Choose at least two problems from the following list. You can do the additional problem for bonus points.
Chapter 6: Exercise 6.3 (verify that the Fourier vectors in C^N form and orthonormal basis).
Chapter 6: Exercise 6.30 (interplay of circular convolution with discrete Fourier transform DFT, and fast convolution).
Chapter 6: Exercise 6.32 (interplay of circular convolution with shift invariance).
Reading Assignment: Sections 6.1-6.4
Homework 4 (due on Friday 2/23/24 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 Friday 2/16/24 at 11:59pm): Choose at least two problems from the following list. You can do the 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, the Cauchy-Schwarz inequality, and the triangle inequality).
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 Friday 2/9/24 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 sums).
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 show they are good kernels).
Reading assignment: Chapter 3 and 4 (go back to Chapter 2 as needed).
Homework 1 (due in Canvas on Tuesday 01/30/24 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.
Let f be a function defined on [-pi, pi) by f(x)=1 if x is in [-pi/2, pi/2) and 0 otherwise, extend the function periodically on R. Calculate the Fourier coefficients and the Fourier series in term of exponentials of f(x). Repeat (i) for f(x):= |sin(x)|, Note that both are even functions, you should be able to write their Fourier series as a purely cosine series, do it. Hint: you can do integration by parts directly on the complex exponential e^{inx}, its antiderivative when n is not 0 is what you would expect e^{inx}/in (why?).
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).
Reading assignment: Chapter 3.
Warmup Homework (due on Sunday 1/21/24 at 11:59pm):
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.
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Last updated: Jan 11, 2024