MATHEMATICS 222 - REAL ANALYSIS

Professor: Jill Pipher
Office: 104 Kassar-Gould House
Phone: (401) 863-3319
Fax: (401) 863-9013
Email: jpipher@math.brown.edu
Office Hours: Wednesday afternoons 1:30-4 and by appointment.


The main text for the course is Folland's Real Analysis. I will maintain a weekly calendar of the topics we cover (see below). The course begins with the basics of point set topology, and continues with functional analysis and $L^p$ and Hilbert spaces. Additional topics from Fourier analysis and geometric measure theory will be considered, as time permits.

GRADING P0LICY. Homework will be assigned regularly and collected periodically. Your final grade will be a combination of the homework grade (50%) and the final exam (50%). The final will be an in-class written exam given on TUESDAY MAY 6, 2:30-5.

HOMEWORK POLICY. I will let you know at least a week in advance of my collecting homework assignments. Working together: You may discuss the problems and "work together" but you must write up your own answers. If you believe that your answer is largely the work of another student, or if you find a major portion of the solution in a textbook, please cite these as resources on your hw papers. Collaboration is good, and helpful. Remember that it is also important to work independently.

LATEST NEWS:

Regular office hours are on Wednesday afternoons. Make an appointment if this time/day doesn't work for you.

April 8, 2008: Most of the material we are doing now is adequately covered in Folland's book, but here are some additional resources on Fourier Series/Fourier Transform for those of you interested in reading more:

1. Dym and McKean, Fourier Series and Integrals.
2. Mark Pinsky, Introduction to Fourier Analysis and Wavelets
3. Terry Tao's lecture notes for Math247A,B at www.math.ucla.edu/~tao

Here are the topics we've covered in distributions:
1. basic definitions, convolutions, Fourier Transform, derivatives of distributions.
2. If sufficiently many (weak) derivatives of a distribution belong to $L^p$ then the distribution has a continuous representative.
3. Bounded linear maps from $L^p$ to $L^q$ which commute with translations are given by convolution with a distribution.
4. The fundamental solution of a linear PDE.



Calendar Material Covered Homework
Jan. 24 Review of measure theory, started point-set topolgy, chapter 4 of Folland. #4.9
Jan. 29 Finish 4.1, and on to 4.2 #4.5, 4.10
Jan. 31 4.2:Continuous functions, Urysohn Lemma. #4.16, 4.17
Feb. 5 Tietze Extension Theorem, Nets (4.2 and 4.3) #4.15 and 4.28(a),(b).
Feb. 7 Compactness, Section 4.4 Problems 4.30, 4.32,
Feb. 12 Tychonoff's theorem, a version of Arzela-Ascoli Problem 4.65
Feb. 14 Begin chapter 5: Banach spaces/linear functionals Problems 5.7, 5.15
Feb. 19 Brown holiday
Feb. 21 Normed vector spaces. section 5.2 5.18, 5.21, 5.22, 5.25(give an example showing the last assertion)
Feb. 26 NO CLASS
Feb. 28 Three big theorems in functional analysis: section 5.3 #5.36, 5.38
Mar. 4 Review of L^p spaces, duality, weak topology #6.20(b), 6.21, 6.22
Mar. 6 Weak and weak* topologies.
Mar. 11 Alaoglu's Theorem, Hilbert spaces #5.47, 5.54, 5.55, 5.56, 5.63
Mar. 13 Hilbert spaces
Mar. 18 Fourier series, completeness of the trig system Why isn't D_N(x) an approximate identity?
Mar. 20 applications of Fourier series (heat equation, random walks)
April 1. The Fourier transform on L^1(R^n), examples.
April 3 Plancherel, the F.T. on L^2
April 8 Riesz-Thorin, Hausdorff-Young, Poisson summation p. 255 #14,22, p. 262 #26,30.
April 10 Poisson summation and applications, sampling.
April 15 no class
April 17 Radon measures, dual of C_0(X)
April 22 Radon measures, duality continued... p.255 #18, p. 225 #22,23,24.
April 24 Tempered Distributions One problem assigned in class
April 29 Tempered distributions No more hw