MATH 601 (Spring 2014)

Instructor's contact information

Michael Robinson
226 Gray Hall
michaelr at american {dot} edu
Office hours:
My research website
Feel free to contact me with any and all questions (course-related or not)

Quick Links

Course description
Homework assignments
Course schedule
Information about exams
Some useful links
Course policies

Course description

Harmonic analysis on the circle, the real line, and on groups. The main concepts are: periodic functions, Fourier series, Fourier transform and spherical harmonics. The course includes a brief account of the necessary ingredients from the theory of the Lebesgue integral. Usually offered alternate springs. Prerequisite: MATH-310, MATH-313, and MATH-503, or permission of instructor.
The overall objectives of this course are to
  1. Introduce students to transform-based methods for analysis and synthesis of functions on specific geometric domains
  2. Teach students to compute transforms of elementary functions
  3. Introduce students to the practical application of these concepts in modern signal processing systems
There is no required book, however much of the spirit of the course is captured in two books:

Homeworks and projects

You must staple your assignments before submission; points will be deducted otherwise! Homeworks are due on the dates listed, to be turned in at the beginning of lecture. If you think that you need more practice, just ask! I'll be happy to give you more problems to work, and will also be happy to answer questions about them.

The programming projects will consist of some preliminary and background theory questions along with a set of guided programming exercises. Instructions and submission of programming projects will be elecronic via the course BlackBoard page! Check there often for updates.

Homework 1: due January 28
Homework 2: due February 6
Project 1: due February 14 (start early!)
Homework 3: due March 18
Homework 4: due March 27
Project 2: due April 8 (start early!)
Homework 5: due April 15
Homework 6: due April 24
Project 3: due April 25 (start early!)

Course schedule

The planned course schedule is below; we will not deviate more than a day in terms of the sections covered. The exams, however, will occur on the days listed below.

Unit 1: Fourier series

After this unit, you should be able to
  1. Prove orthogonality of trigonometric functions (over appropriate intervals) using complex exponentials
  2. Derive the Fourier series decomposition of an arbitrary function
  3. Decompose any function as a sum of an even function and an odd function
  4. Construct Fourier series in general Hilbert spaces using Gram-Schmidt orthonormalization (for instance, spherical harmonic decomposition)
Project 1: Synthesizers for electronic music, due February 14
January 14: Vector spaces, norms, and inner product spaces (Keener 1.1)
January 16: Spectrum of a matrix (Keener 1.2-1.3)
January 21: Least squares solutions (Keener 1.4-1.5)
January 23: Complete vector spaces (Keener 2.1)
January 28: General Fourier series (Keener 2.2)
January 30: Specific Fourier series (Keener 2.2)
February 4: Review for Exam 1
February 6: Exam 1

Unit 2: The Fourier transform

After this unit, you should be able to
  1. Evaluate Fourier transforms of elementary functions
  2. Prove convergence of the Fourier transform under appropriate decay constraints
  3. Explain the construction of the fast Fourier transform for power-of-two length sequences
  4. Explain the Shannon-Nyquist sampling theorem
Project 2: Determining the key signature of a music clip, due April 8
February 11: Complex calculus (Keener 6.1-6.2)
February 13: Contour integration (Keener 6.4)
February 18: The Fourier transform (Keener 7.1-7.2)
February 20: Computing Fourier transforms
February 25: Fourier transforms solve differential equations
February 27: Convergence of the Fourier transform (Keener 7.2)
March 4: Dr. Robinson on travel
March 6: Dr. Robinson on travel
March 18: Inverting the Fourier transform (Keener 7.2.1)
March 20: The Discrete (Fast) Fourier transform
March 25: Review for Exam 2
March 27: Exam 2

Unit 3: Other transforms

After this unit, you should be able to
  1. Compute Laplace transforms of elementary functions
  2. Use Fourier and Laplace transforms to solve ordinary differential equations on unbounded domains
  3. Explain how the poles and zeros of Laplace transforms of operators determine stability
  4. Describe how to fit a specialized transform to a particular domain by diagonalizing the Laplace operator
Project 3: The response of a bowed string, due April 25
April 1: The Laplace transform (Keener 7.3, Oppenheim 9.1)
April 3: Computing Laplace transforms (Oppenheim 9.6)
April 8: Convergence of the Laplace transform (Oppenheim 9.2)
April 10: Inverting the Laplace transform (Oppenheim 9.3)
April 15: Pole-zero plots (Oppenheim 9.4)
April 17: The z-transform (Keener 7.4, Oppenheim 10.1)
April 22: Transforms in context: group representations
April 24: Review for the final exam

Final exam: May 1, 5:30-8:00pm

Course policies


For each exam, you will be permitted to bring one 3"x5" index card with handwritten notes on it. Electronic calculators will not be permitted. However, mechanical calculators (such as this or these) are fine.


You are generally expected to attend the lectures. Please contact me in advance if you cannot attend, especially in the case of an exam.


The course grade will be determined from the following components. You'll accumulate points for each of these over the course of the semester.
20% Homeworks
20% Projects
20% Exam 1
20% Exam 2
20% Final exam

Academic dishonesty

Academic dishonesty is a serious offense. Read our University policies. As applied to this course, you may work together on homeworks and projects, but the work you turn in must be your own. I'm happy to answer any questions you have about these policies.