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2019/2020

Harmonic analysis

Code: 100111 ECTS Credits: 6
Degree Type Year Semester
2500149 Mathematics OT 4 0

Contact

Name:
Artur Nicolau Nos
Email:
Artur.Nicolau@uab.cat

Use of Languages

Principal working language:
catalan (cat)
Some groups entirely in English:
No
Some groups entirely in Catalan:
Yes
Some groups entirely in Spanish:
No

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

  • Assimilate the definition of new mathematical objects, relate them with other contents and deduce their properties.
  • 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 develop the necessary learning skills to undertake further training with a high degree of autonomy.

Learning Outcomes

  1. Generate innovative and competitive proposals for research and professional activities.
  2. 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.
  3. Students must develop the necessary learning skills to undertake further training with a high degree of autonomy.
  4. 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.

Methodology

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

Activities

Title Hours ECTS Learning Outcomes
Type: Directed      
Directed 30 1.2 1, 3, 2
Type: Supervised      
Supervised 20 0.8 4, 1, 3, 2
Type: Autonomous      
Autonomous 85 3.4 4, 1, 3, 2

Assessment

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.

Assessment Activities

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

Bibliography

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

2. Gasquet-Witomski, "Fourier Analysis and applications". Springer-Verlag, 1999.

3. S. Mallat,  "A wavelet tour of signal processing", Academic Press, 1999

4. J.Bruna, Anàlisi Real, Materials UAB, 26.