Math 447, Principles of Analysis II, Spring 2009
Harold P. Boas

Assignments

A standing assignment is to read the relevant sections of the textbook.

Solutions to the following assignments will be available (to students registered in the course) at the TAMU eLearning site. (I am posting the solutions at this password-protected site since the author of the textbook may not want the solutions available to the world.)

Due Wednesday, April 22

• Chapter 14, exercises 1 and 6 on page 218.
• Chapter 14, exercises 10, 11, and 17 on page 220.

Due Wednesday, April 15

Chapter 13, exercises 3, 4, 5, 7, and 11 on page 205.

Due Wednesday, April 8

This assignment is available in pdf.

Due Wednesday, April 1

1. In class, we computed that ∑_n≥1(2/n)sin(nx) is the Fourier series of the odd function on the interval [-π,π] that is equal to (π-x) on [0,π]. Deduce from Parseval's equation that ∑_n≥1(1/n²)=π²/6. (You verified this formula in a different way in the previous assignment).
2. Compute the Fourier series of the absolute-value function |x| on [-π,π] and deduce that the sum of the reciprocals of the squares of the odd positive integers equals π²/8, while the sum of the reciprocals of the fourth powers of the odd positive integers equals π⁴/96.
3. The Dirichlet kernel D_n(t) and the Fejér kernel K_n(t) (see pages 250–256 in the textbook) both peak when t=0. Which value is bigger, D_n(0) or K_n(0)?
4. Compute ||D_n||₂ and ||K_n||₂ (the norms of the Dirichlet and Fejér kernels, using the L₂ norm indicated at the bottom of page 246 in the textbook).
5. Chapter 18, exercise 57 on page 335. [We will use part (c) in a proof later on.]

Due Wednesday, March 25

Chapter 15, exercise 7 on page 250.

Due Wednesday, March 11

The assignment is to study for the midterm exam.

Due Wednesday, March 4

• Chapter 18, exercise 22 on page 322.
• Chapter 18, exercise 23 on page 326.
• Chapter 18, exercises 24 and 26 on page 327.
• Chapter 18, exercise 47 parts (a) and (d) on page 333.

Due Wednesday, February 25

• Chapter 18, exercise 3 on page 316.
• Chapter 18, exercise 4 on page 318.
• Chapter 18, exercise 9 on page 320.
  At this point in the book, the integral has been defined only for nonnegative functions, so you should assume additionally that f is nonnegative. In other words, f(x)≥0 for all x, and f(x)>0 for almost all x. [The statement of the exercise is, however, true in general.]
• Chapter 18, exercises 11 and 12 on page 320.

Due Wednesday, February 18

• Chapter 17, exercise 6 on page 297.
• Chapter 17, exercise 7 on page 298.
• Chapter 17, exercises 25 and 29 on page 303.
• Chapter 17, exercise 30 on page 305.

Due Wednesday, February 11

• Chapter 16, exercises 58 and 60 on page 284.
• Chapter 17, exercises 3 and 4 on page 297.
• Chapter 17, exercise 14 on page 299.

Due Wednesday, February 4

In Chapter 16, do exercises 22, 24, and 28 on page 273 and exercises 42 and 48 on page 282.
Remark: The author attributes exercise 42 to his teacher, William B. Johnson, one of the Distinguished Professors in the Department of Mathematics at Texas A&M.

Due Wednesday, January 28

• In Chapter 16, do exercises 4, 8, 9, and 16 on page 271.
• Additionally, show that properties 1–6 on page 265 in the textbook imply that integration is a linear operation. Since you already know additivity by property 3, what needs to be shown is the multiplicative property that ∫_a^b k f(x) dx = k ∫_a^b f(x) dx for every constant k.
  Remark: Lebesgue says in his famous book Leçons sur l'intégration et la recherche des fonctions primitives that one can prove the multiplicative property "easily" ("sans peine") by using properties 3 and 4.

These pages are copyright © 2009 by Harold P. Boas. All rights reserved.