## Thursday, May 1, 2014

### Final Exam

Take-home exam. Comprehensive, will cover all material learned in class.

Bring the exam to my office MATH 610 between 3:30pm and 5:30pm on Fri, May 9.

### Final Score

Two schemes for calculating your final score will be used: with or without final exam.

Scheme I = (3/10)ME1 + (3/10)ME2 + (1/5)FE + (1/5)HW
Scheme II = (3/8)ME1 + (3/8)ME2 + (1/4)HW

where FE, MEi, HW are the scores (in %) for Final Exam, Midterm Exam i, Homework, respectively.

You will have to email me by 5:30pm on Thur, May 1 to tell whether you will be completing the Final Exam (Scheme I will be used) or not (Scheme II will be used). If you do not contact me by then, your score will be calculated according to Scheme I.

### Midterm Exam 2

Take-home exam. Covers Chap 4-5.

Deadline: Tue, Apr 22, 10:30 am (beginning of class)

### Midterm Exam 1

Take-home exam. Covers Chap 1–3.

(Revised on 2/28:  In problem 3 added missing lim in the third displayed equation)

Note: You will need the password given in class to open this file.

Deadline: Tue, Mar 4, 10:30 am (beginning of class)

## Monday, April 28, 2014

### Course Log

Covered
05/01: Review for Final Exam
04/29:  Ch 6, pp 192-196, Wave equation in $\mathbb{R}^3\times\mathbb{R}$ (finish), Huygens principle, Wave equation in $\mathbb{R}^2\times\mathbb{R}$
- 04/24: Ch 6, pp 187-192, Energy conservation, Wave equation in $\mathbb{R}^3\times\mathbb{R}$ (start)
- 04/22: Overview of Midterm 2, Ch 6, pp 184-187, Wave equation in $\mathbb{R}^d\times\mathbb{R}$
- 04/17: Review for Midterm 2
- 04/15: Ch 6, pp 175-184, Fourier transform in $\mathbb{R}^d$
04/10: Ch 5, pp. 153-161: Poisson summation formula, theta function, heat and Poisson kernels, the Heisenberg uncertainty principle
04/08: Ch 5, pp. 150 -153: Laplace's equation in a halfplane, Poisson kernel, Harmonic functions: mean value property, maximum principle, uniqueness (in bounded and unbounded domains).
04/03: Ch 5, pp. 147-150: Heat equation on $\mathbb{R}$, Laplace's equation in a halfplane, Poisson kernel
04/01: Ch 5, pp. 142-147: Plancherel Formula, Heat equation on $\mathbb{R}$
03/27: Ch 5, pp. 139-142: Gaussian Functions, Fourier Inversion Formula
03/25: Ch 5, pp. 134-138: Fourier transform on the Schwartz space, Gaussian Functions (started)
03/18 - 03/20: Spring break
- 03/13: Overview of Midterm 1
03/11: Ch 4, pp. 118-120: Heat equation on circle, Ch 5, pp. 129-134: Integration on $\mathbb{R}$Definition of Fourier Transform
03/06: Ch 4, pp. 113-118: Continuous nowhere differentiable function
03/04: Ch 4, pp. 105-113: Weyl's equidistribution theorem
- 02/27: Review for Midterm Exam 1
02/25: Ch 4, pp. 100-105: Curves, lengths, and area, Isoperimetric inequality
02/20: Ch 3, pp. 84-87: Counterexample of diverging Fourier series, breaking the symmetry
02/13: Ch 3, pp. 79-84: Mean-square convergence, Parseval's identity, back to pointwise convergence, localization,  Counterexample of diverging Fourier series (start)
02/11: Ch 3, pp. 74-79:  Hilbert and Pre-Hilbert spaces, Best Approximation, Bessel's inequality
02/06: Ch 2, pp. 56-58: Dirichlet problem, Ch 3, pp. 70-74: Review of Vector spaces and inner products.
02/04: Ch 2, pp. 51-56: Cesaro means and summation, Fejer kernel, Abel means and summation, Poisson kernel
- 01/30: Ch 2, pp. 45-51: Convolutions, good kernels
- 01/28: Ch 2, pp. 39-44: Uniqueness of Fourier series
- 01/23: Ch 2, pp. 34-38: Definition of Fourier series, Dirichlet and Poisson kernels.
- 01/21: Ch 1, pp. 18-23: Heat equation, Laplace's equation. Ch 2, pp. 29-33: Riemann integrable functions, functions on unit circle
- 01/16: Ch 1, pp. 11-18: Standing waves, separation of variables, Fourier sine series, Fourier series, plucked string.
- 01/14: Ch 1, pp. 1-11: Simple harmonic motion, derivation of wave equation, traveling waves, D'Alembert's formula

## Tuesday, April 22, 2014

### Homework Assignments

All assignments are from [Stein-Shakarchi]

(Note that there are two types of problems in the textbook: Exercises and Problems)
• #10 Due Tue Apr 29: Chap 6: Exercises 5, 6, 7, 9
• #9 Due Tue, Apr 15: Chap 5: Exercises 13, 15, 16, Problem 4 (skip the proof of part (b))
[Hints for #9][#9 Partial Solutions]
• #8 Due Tue, Apr 8: Chap 5: Exercises 3, 7, 8, 11, 12
[#8 no solutions available]
• #7 Due Thur 03/27:  Chap 4: Exercise 11, Chap 5: Exercises 1, 2, 4, 5
[#7 no solutions available]
• #6 Due Tue 03/11: Chap 4: Exercises 7, 10; Problem 4
[#6 Partial Solutions]
• #5 Due Thur 02/27: Chap 3: Exercises 11, 12, 19, Chap 4: Exercises 1, 4
[#5 no solutions available]
• #4 Extended until Thur 02/20: Chap 3: Exercises 2, 3, 6, 13, 15
[#4 Partial Solutions]
• #3 Due Thur 02/06: Chap 2: Exercises 12, 13(a), 15, 17 (a,b)
[#3 Partial Solutions]
• #2 Due Thur 01/30: Chap 2: Exercises 2, 3, 6, 10, 11
[#2 no solutions available]
• #1 Extended until Tue 01/28 (originally Due Thur 01/23):
Chap 1: Exercises 5, 7, 8, 9, 11; Problem 1.
[#1 Partial Solutions]

## Tuesday, February 11, 2014

### Announcements

- There will be no class on Tue, Feb 18.

## Monday, January 27, 2014

### Course Information

Time and Place: TTh 10:30–11:45 in MATH 215

Instructor: Arshak Petrosyan
Office Hours: TTh 1:30–2:30pm, or by appointment, in MATH 610

Textbook: E.M. Stein & R. Shakarchi, Fourier Analysis: An Introduction, Princeton University Press, 2003.

Syllabus is essentially the first six chapters in [Stein-Shakarchi]:
1. The Genesis of Fourier Analysis
2. Basic Properties of Fourier Series
3. Convergence of Fourier Series
4. Some Applications of Fourier Series
5. The Fourier Transform on R
6. The Fourier Transform on Rd (excluding the higher dimensional wave equation)

Particlular topics include: Fourier series, uniqueness, convolutions, good kernels, Cesaro and Abel summation, Fejer and Poisson kernels, Parseval's identity, Fourier transform, Schwarz class, Gaussian kernels, Plancherel's identity, Poisson summation formula, Radon transform; applications to the wave, heat, and Laplace equations, the isoperimetric inequality, equidistribution theorems.

Homework will be collected weekly on Thursdays. The assignments will be posted on this website at least one week prior to due date.

Exams: There will be two midterm exams and a final exam (project). Exact times will be specified in due course.