# MA49000 Fourier Analysis

Purdue University Spring 2010

## Monday, April 19, 2010

### Midterm Exam 2

Take home exam. Deadline: 11:30am on Fri, Apr 23, 2010

[Midterm Exam 2]
Updated to clarify the statement of problem 1.

### Homework

All assignments are from [Stein-Shakarchi]

(Note that there are two types of problems in the textbook: Exercises and Problems)

#9 Due Fri 04/16: Chap 6: Exercises 1, 2, 5, 7
#8 Due Fri 04/09: Chap 5: Exercises 13, 15, 21, Problem 4 (omit part b).
[Hints][Partial Solutions]
#7 Due Wed 03/31: Chap 5: Exercises 5, 7, 11, 12
#6 Due Fri 03/12: Chap 5: Exercises 1, 2, 3 , 4
#5 Due Mon 02/22: Chap 4: Exercises 1, 4, 7, 10
[Hints]
#4 Due Fri 02/12: Chap 3: Exercises 2, 3, 6, 13, 15
[Partial Solutions]
#3 Due Fri 02/05: Chap 2: Exercises 12, 13(a), 15, 17 (a,b)
[Partial Solutions]
#2 Due Fri 01/29: Chap 2: Exercises 2, 3, 6, 10, 11
#1 Due Fri 01/22: Chap 1: Exercises 5, 7, 8, 9, 11; Problem 1.
[Partial Solutions]

### Course Log

Planned
- Fri 04/23: Ch 6, pp. 204-207: Radon transform: uniqueness and reconstruction.
- Wed 04/21: Ch 6, pp. 201-204: Radon transform in R3
- Mon 04/19: Ch 6, pp. 198-201: Radon transform, X-ray transform in R2.
Covered
- Fri 04/16: Review for Midterm Exam
- Wed 04/14: Ch 6, pp. 196-198: Radial symmetry and Bessel functions.
- Mon 04/12: Ch 6, pp. 175-184: Convolutions, Plancherel formula, Fourier inversion formula
- Fri 04/09: Ch 6, pp. 175-184: Schwartz class. Definition and properties of Fourier transform. Gaussian functions.
- Wed 04/07: Ch 6, pp. 175-184: Integration on Rd. Rotations. Polar (spherical) coordinates.
- Mon 04/05: Ch 5, pp. 158- 161: Heisenberg uncertainty principle
- Fri 04/02: Ch 5, pp. 153-158: Poisson summation formula, theta function, heat and Poisson kernels on the circle.
- Wed 03/31: Ch 5, pp. 152-153: Harmonic functions: mean value property, maximum principle, uniqueness (in bounded and unbounded domains)
- Mon 03/29: Ch 5, pp. 149 -152: Laplace's equation in a halfplane, Poisson kernel
- Fri 03/26: Ch 5, pp. 146-149: Heat equation on R
- Wed 03/24:Ch 5, pp. 144-146: Weierstrass Approx. Theorem, heat equation on R
- Mon 03/22: Ch 5, pp. 142-144: Plancherel Formula
- Mon 03/15 - Fri 03/19: Spring break
- Fri 03/12: Cancelled (because of the Midterm Exam)
- Wed 03/10: Ch 5, pp. 139-142: Gaussian Kernels, Fourier Inversion Formula
- Mon 03/08: Overview of Midterm Exam
- Fri 03/05: Ch 5, pp. 136-139: The Schwartz space (continued), Gaussian Functions.
- Wed 03/03: Ch 5, pp. 134-136: Definition of Fourier Transform, the Schwartz space.
- Mon 03/01: Ch 5, pp. 129-134: Integration on R
- Fri 02/26: Review for Midterm Exam
- Wed 02/24: Ch 4, pp. 116-120: Continuous nowhere differentiable function (continued), Heat equation on circle
- Mon 02/22: Ch 4, pp. 113-116: Continuous nowhere differentiable function
- Fri 02/19: Ch 4, pp. 108-112: Weyl's equidistribution theorem continued.
- Wed 02/17: Ch 4, pp. 104-108: Isoperimetric inequality (finish), Weyl's equidistribution theorem
- Mon 02/15: Ch 4, pp. 100-104: Curves, lengths, and area, Isoperimetric inequality (start)
- Fri 02/12: Ch 3, pp. 85-87: Finish the counterexample of diverging Fourier series.
- Wed 02/10: Ch 3, pp. 84-86: Counterexample of diverging Fourier series, breaking the symmetry
- Mon 02/08: Ch 3, pp. 80-83: Parseval's identity, back to pointwise convergence, localization
- Fri 02/05: Ch 3, pp. 75-80: Hilbert and Pre-Hilbert spaces, mean-square convergence
- Wed 02/03: Ch 3, pp. 70-74: Review of Vector spaces and inner products
- Mon 02/01: Ch 2, pp. 53-58: Abel means and summation, Poisson kernel and Dirichlet problem.
- Fri, 01/29: Ch 2, pp. 48-53: Good kernels, Cesaro means and summation, Fejer kernel.
- Wed, 01/27: Ch 2, pp. 44-48: Convolutions
- Mon, 01/25: Ch 2, pp. 39-44 : Uniqueness of Fourier series
- Fri, 01/22: Ch 2, pp. 29-33: Riemann integrable functions, functions on unit circle, pp. 34-38: Definition of Fourier series, Dirichlet and Poisson kernels.
- Wed, 01/20: Ch 1, pp. 18-23: Heat equation, Laplace's equation.
- Mon, 01/18: no class (MLK day)
- Fri, 01/15: Ch 1, pp. 15-18: Fourier series, plucked string.
- Wed, 01/13: Ch 1, pp. 10-15: D'Alembert's formula, standing waves, separation of variables, Fourier sine series
- Mon, 01/11: Ch 1, pp. 1-10: Simple harmonic motion, derivation of wave equation, traveling waves.

## Monday, March 1, 2010

### Midterm Exam 1

Take home exam. Deadline: 11:30am on Fri, Mar 5, 2010

[Midterm Exam 1]

## Saturday, January 9, 2010

### Course Information

Time and Place: MWF 11:30–12:20 in REC 114

Instructor: Arshak Petrosyan
Office Hours: MWF 10:45–11:30am, 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 Fridays. 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. Exact times will be specified in due course.