18.103 - Fourier Analysis:
Theory and Applications - Spring 2007

| Schedule | (Up) Dates | Homework | Further Reading | Handouts |

General Information

Time MWF, 2 - 3 p.m.
Location Building 4, Room 163
Professor Ben Brubaker (brubaker@math.mit.edu)
Office: 2-267
Office Phone: 3-4079
Office Hours: We 3-4 p.m., Th 2-4 p.m., or by appointment.
TextbookMeasure Theory and Probability, by M. Adams and V. Guillemin
2 Midterms -- 20 % each, Final -- 40%, Homework -- 20%
We will march through Adams and Guillemin, covering Lebesgue measure and integration theory and then moving on to Fourier analysis. Along the way, we'll see how the Lebesgue theory allows for rigorous development of probability (and where Riemann's theory of measure and integration falls short). If time permits, we'll discuss applications of both probability and Fourier analysis to modern analytic number theory.
Prereqs 18.100 is the listed prerequisite. As our starting point will be the theory of Riemann integration and how to develop a more general theory, some familiarity with abstract definitions and proofs will be useful. However, this doesn't rule out 18.100A as a precursor.

Announcements & Dates

  • Important Dates
    • Wednesday, February 7: First Day of Class
    • Monday, February 19: No Class (Presidents Day)
    • Tuesday, February 20: Monday Schedule. Class held.
    • Friday, March 9: Add deadline
    • Wednesday, March 21: First Midterm
    • March 26-30: Spring Vacation
    • Monday, April 16: No Class (Patriot's Day)
    • Wednesday, May 2: Second Midterm
    • Wednesday, May 16: Last Day of Class
    • May 21-25: Final Exam Week
        OUR FINAL: Tuesday, May 22, 1:30-4:30 pm, Walker Gym

Further Reading

  • M. Kac, Statistical Independence in Probability, Analysis, and Number Theory is a classic, and only 96 pages. Adams and Guillemin borrow from this book throughout the beginning sections of Chapter 1.
  • For those interested in learning more about the development of measure theory, including a fuller discussion of Borel sets, check out Measure Theory by Paul Halmos.
  • Wikipedia's entry on the Vitali Set, a construction of a non-measurable set.
  • Royden's book Real Analysis gives a second look at many of the topics we're covering in the course, with special emphasis on the real numbers. The book contains many good exercises and, in fact, major theorems are left to the reader but contain significant hints.
  • J.E. Littlewood is known both as a great mathematician and as a man who could turn a phrase. Here are some links to Littlewood's quotes:
  • I'll try to augment this list with further curiosities as the course gets going.
  • Solutions and Handouts:

  • Second Midterm, with solutions
  • For the Second Midterm, the style will be similar to the first, with 4-5 questions including proofs of two of the following theorems:
    • Product measure (defined by extending measure on "rectangles") agrees with measure defined by iterated integration of characteristic function on slices
    • Fubini's Theorem (for integrable functions)
    • Law of Large Numbers (using expectation values, random variables)
    • L^p(X,u) is a Banach space for all p greater than or equal to 1.
    • L^2(X,u) is a Hilbert space
    • The Fourier coefficients of a complete orthonormal system converge in the Hilbert space norm.
    • The Fourier inversion formula
  • First Midterm, with solutions
  • Instructions and Sample Problems for Midterm I
  • Eventually, I may post further exam solutions and in-class handouts here. For partial solutions to the homework, see the homework page.