MAT 129 HomeWork Page

MAT 129 Homework Page (Spring 2010)

Homework Assignment #0 No due date

E. A. Gonzales-Velasco: "Connections in mathematical analysis: the case of Fourier series," Amer. Math. Monthly, vol.99, no.5, pp.427-441, 1992.

Homework Assignment #1 Due Monday April 5, 2010

Read Chap.1
Do Exercises 1.1.1, 1.1.3, 1.1.4
Here are the actual problems for those who still do not have a textbook.
Solutions

Homework Assignment #2 Due Wednesday April 7, 2010

Read Sec. 2.1
Do Exercises 1.2.5, 1.3.7
Solutions

Homework Assignment #3 Due Friday April 9, 2010

Do Exercises 1.3.3, 1.3.4
Solutions

Homework Assignment #4 Due Monday April 12, 2010

Verify formulas 9, 12, 17 of Table 1 of the textbook and check the smoothness (i.e., differentiability) of the functions lead to the faster decay of the Fourier coefficients.
Read Sec. 2.2
Solutions

Homework Assignment #5 Due Wednesday April 14, 2010

Do Exercises 2.2.1, 2.2.2, 2.2.4
Read Sec. 2.3 and my supplementary course note
Solutions

Homework Assignment #6 Due Friday April 16, 2010

Do Exercises 2.3.3, 2.3.6
Read Sec. 2.4
Solutions

Homework Assignment #7 Due Monday April 19, 2010

Do Exercises 2.3.7, 2.4.3, 2.4.9
Solutions

Homework Assignment #8 Due Wednesday April 21, 2010

Read Sec. 2.6, 3.1.
Do Exercises 2.6.1, 3.1.1
Note: Be careful about the typos in 2.6.1. The equation numbers are incorrect as Prof. Folland pointed out in his errata.
For those who are interested in more about the Gibbs phenomenon, read the following stimulating article: E. Hewitt and R. E. Hewitt: "The Gibbs-Wilbraham phenomenon: An episode in Fourier analysis," Archive for History of Exact Sciences, vol.21, pp.129-160, 1979.
Solutions

Homework Assignment #9 Due Friday April 23, 2010

Read Sec. 3.2
Do Exercises 3.1.5, 3.2.2, 3.2.3
Solutions

Homework Assignment #10 Due Monday April 26, 2010

Read Sec. 3.3
Do Exercises 3.3.2, 3.3.3
Solutions

Homework Assignment #11 Due Wednesday April 28, 2010

Read Sec. 3.3
Do Exercise 3.3.4, 3.3.9
Prove the Cauchy-Schwarz inequality in L²[a,b]. Also, describe the condition that this inequality becomes an equality.
Solutions

Homework Assignment #12 Due Friday April 30, 2010

Read Sec. 3.3
Do Exercises 3.3.6, 3.3.8
Solutions

Homework Assignment #13 Due Monday May 3, 2010

Read Sec. 3.4
Do Exercises 3.3.10 (Problem "a" only), 3.4.2, 3.4.3
For those who got interested in the fact that ∑_n∈N 1/n³ is an irrational number, see the wikipedia page and the references therein as well as this Mathworld page and the references therein.
Solutions

Homework Assignment #14 Due Wednesday May 5, 2010

Read Sec. 3.4
Do Exercises 3.4.6, 3.4.7
Solutions

Homework Assignment #15 Due Friday May 7, 2010

Read Sec. 3.5
Do Exercises 3.5.3, 3.5.7
Read my article, in particular, Corollary 6 and its proof described in Appendix B, which is a good example of the regular Sturm-Liouville problem on the unit interval with unconventional boundary conditions.
Solutions

Homework Assignment #16 Due Monday May 10, 2010

Read Sec. 3.5
Do Exercises 3.5.10, 3.5.12
Solutions

Homework Assignments Due Wednesday May 12 and Friday May 14, 2010

No homework assignments except reading Sec. 4.1 and 4.2

Homework Assignment #17 Due Monday May 17, 2010

Read Sec. 4.1, 4.2
Do Exercise 4.2.1, 4.2.2
Solutions

Homework Assignment #18 Due Wednesday May 19, 2010

Read Sec. 4.2
Do Exercise 4.2.3, 4.2.5
Solutions

Homework Assignment #19 Due Friday May 21, 2010

Do Exercises 4.2.7, 4.2.9
Download the following article and read at least the introduction section: P. Perona and J. Malik: "Scale-space and edge detection using anisotropic diffusion," IEEE Trans. Pattern Anal. Machine Intell., vol.12, no.7, pp.629-639, 1990.
Solutions

Homework Assignment #20 Due Monday May 24, 2010

Read Sec. 4.3
Do Exercises 4.3.3, 4.3.6
Solutions

Homework Assignment #21 Due Wednesday May 26, 2010

Read Sec. 4.4
Do Exercises 4.4.1, 4.4.3
Solutions

Homework Assignment #22 (Last One!) Due Friday May 28, 2010

Do Exercises 4.4.5, 4.4.7
Solutions

Suggested Problems (No need to turn in) posted on Monday May 24, 2010

Read Sec. 7.1, 7.2
Do Exercises 7.1.1, 7.1.2
Solutions

Suggested Problems (No need to turn in) posted on Wednesday May 26, 2010

Do Exercises 7.2.3, 7.2.4, 7.2.6
Solutions

Suggested Problems (No need to turn in) and Supplementary Note posted on Friday May 28, 2010

Do Exercise 7.2.7, 7.3.9 (If you have an older version, i.e., the first three printings of the textbook published by Brooks/Cole, please note that there is a typo in 7.3.9(b) as pointed out by Prof. Folland's errata.)
My supplementary course note on the inverse Fourier transform and the Fourier transform on L²
Solutions

Please email me if you have any comments or questions!
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