Degree | Type | Year | Semester |
---|---|---|---|
2500149 Mathematics | OT | 4 | 2 |
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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
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
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.
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.
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 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.
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 |
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.
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