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Harmonic analysis

Code: 100111 ECTS Credits: 6
2024/2025
Degree Type Year
2500149 Mathematics OT 4

Contact

Name:
Josep Maria Burgues Badia
Email:
josepmaria.burgues@uab.cat

Teaching groups languages

You can view this information at the end of this document.


Prerequisites

The first and second year Analysis courses of the mathematics degree. It is also useful, but not essential, to have followed the course on Real and Functional Analysis

Objectives and Contextualisation

The main objective is to describe the way in which Harmonic Analysis allows to decompose a function as a sum of elementary waves and the applications of this principle

Competences

  • Actively demonstrate high concern for quality when defending or presenting the conclusions of one's work.
  • Assimilate the definition of new mathematical objects, relate them with other contents and deduce their properties.
  • Develop critical thinking and reasoning and know how to communicate it effectively, both in one's own languages and in a third language.
  • Effectively use bibliographies and electronic resources to obtain information.
  • Formulate hypotheses and devise strategies to confirm or reject them.
  • Generate innovative and competitive proposals for research and professional activities.
  • Students must be capable of applying their knowledge to their work or vocation in a professional way and they should have building arguments and problem resolution skills within their area of study.
  • Students must be capable of collecting and interpreting relevant data (usually within their area of study) in order to make statements that reflect social, scientific or ethical relevant issues.
  • Students must develop the necessary learning skills to undertake further training with a high degree of autonomy.
  • Students must have and understand knowledge of an area of study built on the basis of general secondary education, and while it relies on some advanced textbooks it also includes some aspects coming from the forefront of its field of study.

Learning Outcomes

  1. Actively demonstrate high concern for quality when defending or presenting the conclusions of one's work.
  2. Develop critical thinking and reasoning and know how to communicate it effectively, both in one's own languages and in a third language.
  3. Effectively use bibliographies and electronic resources to obtain information.
  4. Formulate conjectures and devise strategies to confirm or reject said conjectures
  5. Generate innovative and competitive proposals for research and professional activities.
  6. Students must be capable of applying their knowledge to their work or vocation in a professional way and they should have building arguments and problem resolution skills within their area of study.
  7. Students must be capable of collecting and interpreting relevant data (usually within their area of study) in order to make statements that reflect social, scientific or ethical relevant issues.
  8. Students must develop the necessary learning skills to undertake further training with a high degree of autonomy.
  9. Students must have and understand knowledge of an area of study built on the basis of general secondary education, and while it relies on some advanced textbooks it also includes some aspects coming from the forefront of its field of study.
  10. Understand and know how to reproduce basic results in relation to the Hilbert transform.

Content

1. Fourier series and applications
										
											
										
											2. Fourier integrals and applications.
										
											
										
											3. The Poisson summation formula. The Heisenberg Uncertainty Principle
										
											
										
											4. Fourier analysis in finite abelian groups. Dirichlet's theorem on prime numbers in arithmetic progressions.

5. Theory of distributions. Fourier transform of tempered distributions. Applications.

Activities and Methodology

Title Hours ECTS Learning Outcomes
Type: Directed      
Directed 30 1.2 5, 8, 6
Type: Supervised      
Supervised 20 0.8 10, 5, 8, 6
Type: Autonomous      
Autonomous 85 3.4 10, 5, 8, 6

The standard one in Mathematics. Discussion of definitions, examples and Theorems. We will also have problem sessions.

Annotation: Within the schedule set by the centre or degree programme, 15 minutes of one class will be reserved for students to evaluate their lecturers and their courses or modules through questionnaires.


Assessment

Continous Assessment Activities

Title Weighting Hours ECTS Learning Outcomes
Exercises 40% 1 0.04 1, 2, 10, 4, 5, 9, 8, 6, 7, 3
Final Exam 50% 4 0.16 1, 2, 10, 4, 5, 9, 8, 6, 7, 3
Oral Exam 10% 10 0.4 1, 2, 10, 4, 5, 9, 8, 6, 7, 3 

The subject will be evaluated according to the three activities and their weights shown in the table. 
										
											
										
											Students who do not pass the course can repeat the final exam with the same weight.

THERE IS ALSO THE POSSIBILITY OF DOING ONLY ONE FINAL EXAM WITH THAT WEIGHTS THE 100% OF THE FINAL GRADE.

Bibliography

1. E. Stein and R. Shakarchi, "Fourier Analysis, an introduction",  Princeton Lectures in Analysis, Priceton Univresity Press 2007

2. L. Grafakos, "Classical Fourier Analysis", Springer-Verlag,

3. C. Pereyra and L. Ward, "Harmonic Analysis; from Fourier to Wavelets", 2012.

4. R. Strichartz, "A Guide to Distribution Theory and Fourier Transforms". CRC Press, Boca Ratón, FL, 1994.

5. A. H. Zemanian, "Distribution Theory and Transform Analysis: An Introduction to Generalized Functions, with Applications, reprint edition". Dover Publications, New York, 1987.

 

Bibliografia complementaria:

-Geometric Harmonic Analysis I, A Sharp Divergence Theorem with Nontangential Pointwise Traces. D. Mitrea, I. Mitrea and M. Mitrea. Springer-Verlag, 2022.

-Geometric Harmonic Analysis II, Function Spaces Measuring Size and Smoothness on Rough Sets. D. Mitrea, I. Mitrea and M. Mitrea. Springer-Verlag, 2022.

-Geometric Harmonic Analysis III, Integral Representations, Calderón-Zygmund Theory, Fatou Theorems, and Applications to Scattering. D. Mitrea, I. Mitrea and M. Mitrea. Springer-Verlag, 2022.

-Geometric Harmonic Analysis IV, Boundary Layer Potentials in Uniformly Rectifiable Domains, and Applications to Complex Analysis. D. Mitrea, I. Mitrea and M. Mitrea. Springer-Verlag, 2022.




 

 

 


Software

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Language list

Name Group Language Semester Turn
(PAUL) Classroom practices 1 Catalan second semester morning-mixed
(SEM) Seminars 1 Catalan second semester afternoon
(TE) Theory 1 Catalan second semester morning-mixed