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 |
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Lecture 1 |
Introduction. Diagonalizing the second derivative Unit Circle S^1 as a group. Characters.
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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. |
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 |
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13/01 |
Written Exam |
Schnelli | Time: 08.00-12.00 am. Location: Q21 |
16/01, 17/01 |
Oral Exam |
Schnelli |
Time: TBA |
16/04 |
Written Re-Exam |
Schnelli |
Time: 08.00-12.00 am. |
TBA |
Oral Re-Exam |
Schnelli |
Time: TBA. |