Fourier Analysis: Spring 2010

This schedule is tentative and may be modified as necessary. I have marked with an asterisk those (few) topics not covered by Stein-Shakarchi. Please check back each week for homework and read the relevant textbook sections before the lecture

Approximate Schedule for Math W4032 Spring 2010

Date

Lecture

Reading

Homework

  Jan. 20

  § Fourier series: definition, examples and uniqueness

  § Chapter 2

#1 Due 01/27 at 6pm

Chapter 2: Exercises: 1, 2, 4, 6, 7, 8
Chapter 2: Problems: 2
Extra Knowledge (no credit): Exercises: 5, 9; Problems: 1

  Jan. 25, 27

  § Decay of Fourier coefficients
  § Partial sum and convergence
  § Cesaro and Abel summability
  § Convolutions
  § Dirichlet and Poisson kernels

  § Chapter 2

#2 Due 02/03 at 6pm

Chapter 2: Exercises: 11, 13, 14(b), 15
Chapter 3: Exercises: 15
Chapter 2: Problems: 3 (i), (ii), (iii)

  Feb. 1, 3

  § Convolutions and Fourier Series
  § Approximation of identity
  § The Weierstrass approximation theorem
  § Application: the Dirichlet's problem in the unit disc.

  § Chapter 2

#3 Due 02/10 at 6pm

Chapter 2: Exercises: 17, 18, 19, 20
Chapter 1: Exercises: 11
Extra Knowledge (no credit): Chapter 1: Exercises: 6, 8, 10; Problems: 1

  Feb. 8, 10

  § Convergence of Fourier series
  § Mean-square convergence: L^{2} theory
  § Parseval's identity; Bessel's inequality
  § Riemann-Lebesgue lemma
  § Convergence of Fourier series: pointwise convergence
* Dini's criterion; Jordan's criterion

  § Chapter 3

#4 Due 02/15 at 6pm

Chapter 3: Exercises: 9, 11, 16, 18
Extra Knowledge (no credit): Chapter 3: Exercises: 6, 7; Problems: 1, 2

  Feb. 15

* Gibbs phenomenon
  § Riemann localization principle
  § A continuous function with diverging Fourier series

Chapter 3

  Feb. 17

Midterm

The midterm will cover the material of chapters 2 & 3.

In order to prepare for this midterm you should redo the homework sets # 1, # 2, # 3, # 4.

  Feb. 22, 24

  § Some applications of Fourier series
  § The isoperimetric inequality
  * The Poincare and Wirtinger inequalities
  * Monotonicity properties of harmonic functions
  § Weyl's equidistribution theorem


  § Chapter 4

#5 Due 02/26 at 4pm

Chapter 3: Exercises: 12
Chapter 4: Problems: 5
Extra Knowledge (no credit): Chapter 3: Exercises: 17, 19; Chapter 4: Exercises: 1, 2, 3, 4; Problems: 6

  Mar. 1, 3

  § Some applications of Fourier series continued
  § A continuous but nowhere differentiable function
  § The heat equation on the circle.

  § Chapter 4

#6 Due 03/05 at 6pm

Chapter 4: Exercises: 5, 8, 9, 10
Extra Knowledge (no credit): Chapter 4: Exercises: 6; Problems: 2, 3

  Mar. 8, 10

  § The Fourier transform on the real line
  § The Schwartz space and Fourier transform
  § The Fourier inversion
* Eigenfunctions for the Fourier transform in L^{2}(R)

  § Chapter 5

  Mar. 22, 24

  § The Plancherel formula
  § Applications to some PDEs: the heat equation on the real line; the Laplace equation in the upper half-plane

  § Chapter 5

#7 Due 03/24 at 6pm

Chapter 5: Exercises: 1, 2, 3, 5, 7
Extra Knowledge (no credit): Chapter 5: Exercises: 4, 6; Chapter 5: Problems: 7

  Mar. 29, 31

  § The Poisson summation formula.
  § The theta and zeta functions
  § The Heisenberg uncertainty principle.

  § Chapter 5

#8 Due 03/31 at 6pm

Chapter 5: Exercises: 8, 9, 11
Chapter 5: Problems: 4
Extra Knowledge (no credit): Chapter 5: Exercises: 12;
Chapter 5: Problems: 1, 3, 5

  April. 5, 7

  § The Fourier transform in higher dimensions
  § The Schwartz space and Fourier transform
  § Applications to the wave equations

  § Chapter 5

 #9 Due 04/07 at 6pm

Chapter 5: Exercises: 14, 15, 16, 19
Extra Knowledge (no credit): Chapter 5: Exercises: 17, 18;
Chapter 5: Problems: 6, 8

  April. 12, 14

  § Applications to the wave equations continued

  § Chapter 6

#10 Due 04/14 at 6pm

Chapter 5: Exercises: 23
Chapter 6: Exercises: 5, 6, 7
Extra Knowledge (no credit): Chapter 6: Exercises: 1, 2, 3

  April. 19, 21

  § The Radon transform and some of its applications .
  § Finite Fourier Analysis
  § Fourier Analysis on Z(N)

  § Chapter 6

#11 Due 04/28 at 6pm

Chapter 6: Exercises: 8, 12, 13
Chapter 6: Problems: 3
Extra Knowledge (no credit): Chapter 6: Exercises: 14, 15
Chapter 6: Problems: 7, 8

  April. 26, 28

  § The Fast Fourier Transform.
  § Fourier Analysis on finite abelian groups.
  § Characters; Fourier inversion and Plancherel formula.

  § Chapter 7

#12 Due 05/03 at 6pm

Chapter 7: Exercises: 2, 3, 4, 8, 9 (a), 13
Extra Knowledge (no credit): Chapter 7: Exercises: 9 b, 10, 11, 12

  May 3

REVIEW

 

  May 12

FINAL EXAM