Math 414-501 — Spring 2009

Assignment 1

• Problems: §1.4: 1, 3-7. (You may use any software to make the required plots.)

Assignment 2

• Read sections 1.3.1-1.3.4
• Problems:
  1. §1.4: 8, (You may use any software to make the required plots.), 11, 16
  2. Let f(x) = e^2x, - π &le x ≤ π. Find the complex form of the Fourier series for f.
  3. Let g(x) = e^2x, 0 &le x ≤ 2π. Find the complex form of the Fourier series for g. Notice the interval is [0,2π] in this problem, which is different from the previous problem.
  4. Let u(x) and v(x) be be continuously differentiable 2π periodic functions that the complex Fourier series
     u(x) = ∑_n α_n e^inx and v(x) = ∑_n β_n e^inx.
     Define the convolution of u and v to be u∗v(x) := (2π)^-1 ∫_-π^πu(t) v(x − t) dt. Show that the complex form of the Fourier series for u∗v is
     u∗v(x) = ∑_n α_n β_n e^inx.
     Hint: Use Theorem 1.18 in the text directly on u∗v to get a double integral for the coefficient, and then intechange the two integrals.

Due Friday, February 6.

Assignment 3

• Read sections 0.1-0.4 (innper products), 1.3.5, and 2.1
• Problems:
  1. §1.4: 18, 19(b,d), 28
  2. § 0.8: 2, 3, 4

Due Friday, February 13.

Assignment 4

• Read sections 0.5 (inner products), 1.3.5, and 2.1
• Problems:
  1. §1.4: 17, 20(b,d)
  2. This exercise involves modifying Theorem 0.21, p. 30.
     1. Suppose that the basis vectors e_j in Theorem 0.21 are not orthonormal. Show that the minimizer coefficients α_j satisfy the matrix equation equation Aα = d, where d_j = < v,e_j> and A_j,k = < e_j,e_k>.
     2. Suppose that the inner product space is L²[0,1]. Find the minimizer for e^x if the basis for the subspace is {1, x, x²}. Use this basis; don't orthonormalize. (This is the least-squares quadratic fit to e^x.)
     3. Plot the quadratic minimizer and e^x on 0 ≤ x ≤ 1.
  3. Consider Fourier series for the 2π periodic extension of f(x) = x in Example 1.29. Estimate the "energy" in the residual signal r_N(x) := f(x) − f_N(x). (Hint: In class we showed that ||r_N||² = ||f||² − ||f_N||². Use this, the sum of the series in Exercise 1.18, and the trick we did in class to estimate the tail end of a series.)

Due Friday, February 20.

Assignment 5

• Read sections 2.1 and 2.2.
• Problems
  1. §2.6: 1, 2, 4, 7
  2. Find the Fourier transform of f(t) = e^-|t|. In addition, use this transform and the properties listed in Theorem 2.6 to find the Fourier transforms of the following functions:
     1. t e^-|t|
     2. e^-2|t-3|
     3. sign(t)e^-|t| (Hint : differentiate e^-|t|.)
     4. 1/(1+(t-2)²)  (Hint: How are Fourier transforms and inverse Fourier transforms related?)

  3. Find the Fouirer transforms of these functions.
     1. g(t) = 1 if a < t < b and g(t) = 0 otherwise.

     2. h(t) = -1 if -a < t < 0, h(t) = 1 if 0 < t < a, and h(t) = 0 otherwise.

Due Friday, 6 March

Assignment 6

• Read sections 2.3 and 2.4.
• Problems
  1. §2.6: 5, 6

  2. §2.6: 12. This problem involves the averaging filter discussed in class. Use tables to do the integrals involved. Plots are required.

  3. Use the Plancheral/Parseval Theorem 2.12 and Ex. 1, §2.6, to find this integral:

     ∫_ℝλ²sin²(πλ) (λ² − 9)⁻² dλ

Due Friday, 13 March

Assignment 7

• Read sections 2.5 and 3.1.
• Problems. §2.6: 8, 10, 11, 14

Due Friday, 27 March

Assignment 8

• Read sections 3.1 and 3.2.
• Problems.
  1. Let s > 0 and suppose f_s is the Gaussian in problem 6, p. 140. Calculate the uncertainty product in Theorem 2.26, p. 136, provided both a = 0 and α = 0. (You may use the fourier transform of f_s found in problem 6.)
  2. §3.1: 2
  3. §3.1: 14(a,b)
  4. For M₄ in 14(b), show that the columns of F₄ (cf. p. 148, (3.2) with n = 4) are eigenvectors of M₄.
  5. Use Matlab or some other software to find the FFT for the functions below. All functions are defined and periodic on [0,2π]. In each case, plot the absolute value of the FFT.
     1. f(t) = t(2π − t), n = 64.
     2. f(t) = e^−t, n = 128.
     3. f(t) = t, n = 256.

Due Friday, 3 April.

Assignment 9

• Read chapter 4
• Problems.
  1. Section 4.5: 1, 2, 6, 7

Due Friday, 24 April.

Assignment 10

• Read sections 5.1, 5.2, 5.3.3, 5.3.4
• Problems.
  1. Section 5.4: 1(b), 2(a,c), 3, 4(b), 5, 8(c,d,e,f), 11

Due Friday, 1 May.