Course plan and reading suggestions

This course has two modules:

Module Blue:  Fourier series

Module Yellow:  Fourier transform.

Schedule is found here Links to an external site..

Datum, Place Topic Teacher Reading suggestions

 

Module Blue

Lecture 1

Introduction.

Diagonalizing the second derivative

Unit Circle S^1 as a group. Characters. 

 

Duits

4.1

4.2

1.3

Lecture 2

Fourier series of continuous and smooth functions. Fejer and Dirichlet kernels

Duits

1.4

Lecture 3

L^2(S^1) as a Hilbert space. 

Properties of Fourier series in L^2(S^1)

Duits

1.2

1.3

Lecture 4

Fourierseries of L^1(S^1) functions. Convolution and Fourier series

Duits 1.5
Lecture 5

Application: isoperimetrical inequality. Polynomial approximation.

Duits

1.7

Lecture 6

Fourierseries and probability. Z-valued random variables. Characteristic function. CLT Duits 1.10
Lecture 7

Fourierseries with several variables. Random walks in d dimensions

Duits 1.10
Lecture 8

One-dimensional heatequation.

Duits  1.8
Module Yellow
Lecture 9 Fourier integrals on Schwartz space Schnelli 2.2
Lecture 10 Fourier transform on L2(R), Plancherel theorem, Hermite functions.  Schnelli  2.3, 2.5
Lecture 11

Fourier integrals on L1(R), Riemann-Lebesque lemma.
Applications: Airy equation, Poisson summation formula, Heisenberg uncertainty

Schnelli

2.6

 

2.7, 2.8

Lecture 12

Fourier transform in several dimensions: wave equation in Rd

Recap of complex analysis.

Schnelli

2.10, 2.11(1)

 

3.1,

Lecture 13

Fourier transform and analyticity: Hardy Theorem, Paley-Wiener Theorem.

Fourier Analysis on commutative groups. 

Schnelli

3.2, 3.3

4.4, 4.5

Lecture 14 Applications to Number Theory: Zeta function and Euler Products, Dirichlet's Theorem. Schnelli 8.1 in Stein-Shakarchi
Lecutre 15 Applications to Number Theory: Dirichlet characters and L-functions. Schnelli 8.2 in Stein-Shakarchi
Lecture 16 Proof of Dirichlet's Theorem. Schnelli 8.3 in Stein-Shakarchi (``Case II: Real Dirichlet Characters'' is optional).

Examinations 

 

 13/01

Written Exam

Schnelli Time: 08.00-12.00 am.
Location: Q21

 16/01, 17/01

Oral Exam

Schnelli

Time: TBA
Location: TBA

16/04

Written Re-Exam

Schnelli

Time: 08.00-12.00 am.
Location: Q13. 

TBA

Oral Re-Exam

Schnelli

Time: TBA.
Location: TBA.