Course in Harmonic Analysis
Diploma de Estudos Avançados em Matemática - Instituto Superior Técnico de Lisboa
2024 / 25
Diploma de Estudos Avançados em Matemática - Instituto Superior Técnico de Lisboa
2024 / 25
Welcome! I am the Responsible Teacher for this course. You can contact me via email at giuseppe.negro@tecnico.ulisboa.pt. The official language will be English, but you can also communicate with me in Portuguese, Italian, French or Spanish, if needed.
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Objectives. Harmonic Analysis is a loose term. It describes a family of mathematical methods and disciplines with one thing in common: the concept of decomposing functions exploiting symmetry. We will study a selection of results, starting with the analysis of finite signals, and progress our way to the continuous Fourier analysis of the Euclidean space. We will pay special attention to the concept of Fourier Restriction. We will try to prioritize modern results and research literature over the more traditional books.
Prerequisites. These are kept to a minimum. Linear algebra and multi-variable calculus are essential. We will need some basic functional analysis and measure theory. If these are lacking I will give some minimal complementary material. Most importantly we will need a good dose of mathematical maturity, in order to navigate through modern research literature.
Grading. I will distribute some homework. These will account for roughly half of the grade. At the end of the course there will be an oral exam.
Timetable. Please check the official webpage on Fénix.
Bibliography. I will suggest bibliographical material on a weekly basis (roughly).
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Week 1. Welcome! We will start by reading Chapter 7, "Finite Fourier Analysis", from the book
Stein, E.M. and Shakarchi, R., "Fourier analysis. An introduction", Princeton University Press.
During Lecture 2 we took a detour to study some applications to image processing: linear filters and the JPEG format.
--->I strongly recommend watching the video But what is a convolution? by 3Blue1Brown. This video covers several topics we touched this week, such as linear filters in image processing and the relation between convolution and the fast Fourier transform. <---
Bibliography on image processing:
F. Chamizo, Aplicaciones del análisis armónico, lecture notes, UAM Madrid (in Spanish). See §1.3.3.
For more information, check the book
F. Chamizo, A course in signal processing, UAM.
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Week 2. Plan for this week:
1) Finish our discussion of the JPEG format.
2) Quickly discuss the case of general finite abelian groups.
3) Begin our study of the paper
A. Iosevich and A. Mayeli. "Uncertainty principles, restriction, Bourgain's Lambda_q theorem and signal recovery". Appl. Comput. Harmon. A. 76 (2025).
https://doi.org/10.1016/j.acha.2024.101734
If you have trouble downloading the paper via the DOI link above, you can download it at A.Iosevich's webpage via the following link:
https://people.math.rochester.edu/faculty/iosevich/im.pdf
During Lecture 4 we discussed: Section 2 (whole) and Proposition 3.1, Remark 3.3 and Definition 3.4 from Iosevich - Mayeli.
Exercise Sheet 1 (due March 16th).
ATTENTION. This is the solution to the first exercise of the first sheet.
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Week 3. We will continue our exploration of the concept of "Fourier Restriction" in the discrete setting, and its applications to signal recovery. We will explore parts of the influential paper
G. Mockenhaupt, T. Tao, "Restriction and Kakeya phenomena for finite fields", Duke Math. J. 121(1): 35-74 (2014).
https://doi.org/10.1215/S0012-7094-04-12112-8
If you cannot access the former, use the arXiv version at https://arxiv.org/pdf/math/0204234
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Week 4. This week we will finish our exploration of Mockenhaupt--Tao. I recommend reading pp. 1--5 of Tony Carbery's lecture notes at https://www.maths.ed.ac.uk/~carbery/analysis/notes/fflpublic.pdf.
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END OF PART 1. Fourier Analysis on Finite Abelian Groups.
Topics covered:
Stein -- Shakarchi chapter 7.
F. Chamizo lecture notes: linear filters and the JPEG algorithm.
Iosevich--Mayeli: the finite uncertainty principle and consequences for signal recovery (uniqueness of recovery and the DRA algorithm). Restriction inequalities and consequences: new uncertainty principles, new signal recovery theorems.
Mockenhaupt--Tao: on the paraboloid, the constant R*(2->2k) is bounded independently of the base field.
Extra: construction of the field of order p^2. If you missed the class, see https://en.wikipedia.org/wiki/Finite_field#GF(p2)_for_an_odd_prime_p
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PART 2. Examples of Fourier Analysis on compact groups.
Week 5. The starting point will be Lecture 3 from the lecture notes:
H. Cohn. Packing, coding and ground states. https://arxiv.org/abs/1603.05202
For preliminaries on Hilbert space theory, see Lieb & Loss "Analysis", 2nd ed., Section 2.21.
The proof of completeness of the trigonometric system is taken from Rudin, "Real and complex analysis", section 4.24.
If you missed Thursday 20 March class (bad weather): please study pp. 27--29 of H. Cohn's lecture notes.
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Week 6. We continue our investigation of spherical harmonics.
Lecture 25 March - recap on representation theory
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Week 7. We conclude our investigation of spherical harmonics.
END OF PART 2: Examples of Fourier Analysis on compact groups (abelian and nonabelian).
Topics covered:
Basics of Hilbert spaces (Lieb & Loss, "Analysis", 2nd edition, §2.21).
Fourier series on tori. Basics of representation theory. Spherical harmonics as homogeneous harmonic polynomials, or as eigenfunctions of the Laplace--Beltrami. The reproducing kernels are ultraspherical polynomials. Consequence: spaces of spherical harmonics of degree n are irreps. (H. Cohn, Packing, coding and ground states. https://arxiv.org/abs/1603.05202 , Lecture 3 full. Lecture 4, §5).
More spherical harmonics: The addition formula. A formula for the orthogonal projector onto the space of spherical harmonics of degree n. The Legendre harmonic. Spherical convolutions and the Funk--Hecke theorem. (C. Müller, Spherical harmonics, Springer 1966. pp. 8--11 and 18--20).
Warning: H. Cohn proved the density of trig polynomials in L²(S¹) and of general polynomials in L²(S^(d-1)) via the Stone--Weierstrass theorem. Instead, we followed:
W. Rudin, Real and complex analysis, 3rd ed, §4.24 - trigonometric polynomials on S¹
K. Atkinson, W. Han. Spherical harmonics and approximations on the unit sphere. Springer 2012. §2.8.1 - polynomials on S^(d-1).
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Week 8. Start of Part 3: Harmonic Analysis on Euclidean space.
We follow the book:
E. Stein, G. Weiss, Introduction to Fourier Analysis on Euclidean spaces. Princeton University Press (1971).
During April 29 we read pp.1--5.
Week 9-10. We continue our reading of Stein--Weiss, chapter 1.
We did not prove the log-convexity of Lp norms. I recommend reading Terry Tao's blog, entry "245C: Interpolation of Lp spaces", Lemma 9.
Final weeks. We finished reading Stein--Weiss chapter 1, including tempered distributions and the fundamental theorem on translation--invariant operators on $L^p(\mathbb R^d)$. (Theorem 3.16).
The final topic is the paper D. Foschi, "Global maximizers for the sphere adjoint Fourier restricion inequality", JFA 2015 (https://www.sciencedirect.com/science/article/pii/S0022123614004431).
Read slides 3, 4, 5 and 6 of my slides. I will give a Zoom class on Thursday, 5 June to explain the final part of the paper.
Final Homework: Please hand it in before the oral exam. (Sheet 4).