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MATH 857: Topics in Analysis: Fourier Analysis and Convexity (Spring 2017)

Course Information:

Syllabus

Results on the Convergence of Fourier Series

References:

[B] Ball, K. An Elementary introduction to Modern Convex Geometry (in the book "Flavors of Geometry", edited by S. Levi)
[D] Duoandikoetxea: Fourier Analysis
[F] Folland, G: Real Analysis
[Ka] Katznelson, Y: An introduction to harmonic analysis
[K] Koldobsky, A: Fourier Analysis in Convex Geometry
[KY] Koldobsky, A and Yaskin, V: The interface between Convex Geometry and Harmonic Analysis
[PW] Pereyra, C and Ward, L: Harmonic Analysis: From Fourier to Wavelets
[Sc] Schneider, R: The use of spherical harmonics in convex geometry.
[SS] Stein, E and Shakarchi, R: Fourier Analysis: An introduction

Class presentations:

1. The first class presentations will start on February 6. There are three projects, to be completed in groups of 2. You have to submit a write-up in LaTeX of your project, and present to the class. I will be out of town the week before, and you can use the class time to work on this project.
   

• Solving the wave equation (including Ex 9 on page 27 and Ex 3 on page 59) (Liz and Mike P.)
  
• Solving the heat equation on the disc (including Ex 10 and 11 on page 27 and Ex 18 on page 64) (Syed and Gulnar)
  
• A continuous function whose Fourier series is divergent at a point (including Ex 8 on page 60)   (Mike M. and Chase)
  

HW 1 (Due Friday February 24):  Click here.
HW 2 (Due March 29): Click here.

CLASS DIARY:     

         Week of Jan 23. On Monday we covered pages 31-34 in [PW] (discussion about the nested spaces of functions where we are working) and 68-73 in [PW] (the problem of the
                                     convergence of partial sums of Fourier series, with a list of the main positive results and counterexamples - a survey of those can be found here.)
                                     On Wednesday we proved  Theorem 2.1 and Corollaries 2.2, 2.3, 2.4 in [SS] (two of these appear also in [PW], Thms 3.6 and 3.7).
                                     On Friday we covered Convolutions and the Dirichlet kernel ([SS], section 2.3 or [PW], sections 4.1 and 4.2).

         Week of Jan 30. I will be out of town.

         Week of Feb 6.  Presentations.

         Week of Feb 13.  Presentations on Monday and Wednesday. On Friday we covered the theorem about Good Kernels and Cesaro convergence ([SS] Sections 2.4 and 2.5, or [PW]
                                      Sections 4.3 and 4.4).

         Week of Feb 20. On Wednesday we recapped all the previous material, and introduced the Fourier transform ([PW] pages 162-164).

         Week of Feb 27. Fourier transform on the Schwartz class ([SS] Chapter 5, pages 134-144, for the 1-dim Fourier transform, and Chapter 6, pages 175-184 for the n-dim).
                                     Or [PW], Chapter 7.

         Week of March 6: Intro to Convex Geometry and basic concepts [K] Chapter 1, Section 2.1
       
         Week of March 13: Spring Break

         Week of March 20: Brunn-Minkovski and some consequences, [K] Section 2.2 (up to page 21). Intro to concentration of measure, [B] Lecture 1.

         Week of March 27: Fourier transform of distributions, [K] Section 2.5 (See also [PW], Chapter 8, which is very friendly written).

         Week of April 3: Hyperplane sections of Lp balls, [K] Section 3.1. Keith Ball's counterexample for Busemann-Petty (capstone project notes).

         Week of April 10: Radon transform and Fourier transform [K] Page 27 and Section 3.2. Results from Section 3.3 (without proofs).

         Week of April 17: 

         Week of April 24:

         Week of May 1: