Course plan and reading suggestions
This course has two modules:
Module I: Fourier series
Module II: Fourier transform.
Due to COVID-19 all lecture for P1 will be online using zoom. The zoom link is https://kth-se.zoom.us/j/66304938297 Links to an external site.
The lectures for P2 will be on campus but in hybrid mode, so you will be able to follow the lecture online using the above zoom link.!
| Date and Place | Location | Topic | Reading suggestions | Notes |
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Module I |
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(The notes are not proofread nor prepared in advance. These are writings during the lectures and may contain errors. Always check the book in case of confusion) | ||
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Lecture 1 Tuesday 31/8 Time: 15-17 |
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Introduction. Diagonalizing the second derivative Unit Circle S^1 as a group. Characters.
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4.1 4.2 1.3 |
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Lecture 2 Tuesday 7/9 Time: 15-17 |
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Fourier series of continuous and smooth functions. Fejer and Dirichlet kernels |
1.4 |
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Lecture 3 Friday 10/9 Time: 13-15 |
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L^2(S^1) as a Hilbert space. Properties of Fourier series in L^2(S^1) |
1.2 1.3 |
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Lecture 4 Tuesday 14/9 Time: 15-17 |
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Properties of Fourier series in L^2(S^1)
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1.4 | Note 14 Sep 2021.pdf Download Note 14 Sep 2021.pdf |
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Lecture 5 Tuesday 21/9 Time: 15-17 |
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Fourierseries of L^1(S^1) functions. Convolution and Fourier series |
1.5 |
Note 21 Sep 2021.pdf Download Note 21 Sep 2021.pdf An overview of convergence results: |
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Lecture 6 Tuesday 28/9 Time: 15-17 |
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Application: isoperimetrical inequality. Polynomial approximation. | 1.7 | Note 28 Sep 2021 (2).pdf Download Note 28 Sep 2021 (2).pdf |
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Lecture 7 Tuesday 5/10 Time: 15-17 |
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One-dimensional heat equation. |
1.8 | Note 5 Oct 2021.pdf Download Note 5 Oct 2021.pdf |
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Lecture 8 Tuesday 19/10 (change in schedule!) Time: 15-17 |
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Fourierseries with several variables. Random walks in d dimensions |
1.10 | Note 19 Oct 2021.pdf Download Note 19 Oct 2021.pdf |
| Module II | (PRELIMINARY) | |||
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Lecture 9
Tuesday 2/11 Time: 10-12
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E52 |
Fourier integrals on Schwartz space | 2.2 | Note 2 Nov 2021.pdf Download Note 2 Nov 2021.pdf |
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Lecture 10 Friday 5/10 Time: 13-15 |
E31 |
Fourier transform on L2(R), Plancherel theorem, Hermite functions. | 2.3, 2.5 | Note 5 Nov 2021.pdf Download Note 5 Nov 2021.pdf |
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Lecture 11 Tuesday 9/11 Time: 10-12 |
E35 |
Fourier integrals on L1(R), Riemann-Lebesque lemma. |
2.6
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Lecture 12 Tuesday 16/11 Time: 10-12 |
E53 |
Applications: ODE, heat equation. Heisenberg inequality |
2.7, 2.8 |
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Lecture 13 Tuesday 23/11 Time: 10-12 |
E53 |
Poisson summation formula, Jacobi theta identity Minkowski's Theorem.
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2.11 |
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Lecture 14 Tuesday 30/11 Time: 10-12 |
Q26 |
Bessel transform, Wave equation in dimension 2 and 3. Recap of complex function theory |
2.10, 2.11 3.1 |
Note 30 Nov 2021.pdf Download Note 30 Nov 2021.pdf |
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Lecutre 15 Tuesday 7/12 Time: 10-12 |
E53 |
Complex function theory, Phragmen-Lindelöf, Hardy's theorem,
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3.1 3.2 |
Note 7 Dec 2021.pdf Download Note 7 Dec 2021.pdf |
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Lecture 16 Friday 12/11 Time: 10-12 |
D34 |
Paley Wiener space. Airy equation (extra material) |
3.3 | Note 10 Dec 2021-1.pdf Download Note 10 Dec 2021-1.pdf |
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Examination |
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12/01 13/01 14/01 |
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Oral exam | ||
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