Topics of Advanced Analysis

Objectives

By the end of this course, the student should have acquired knowledge, skills and competences in order to:

-Use fluently the elementary properties, results and procedures concerning distributions, in view of applications to differential equations;

-Understand the topology of the distribution''s space and apply it to the different notions of convergence of a sequence of functions.

-Differentiate distributions and establish the relationship between this type of differentiation and differentiation in the classical sense.

-To be familiar with the notion of differential operator and fundamental solution.

-Apply the Fourier Transform to tempered distributions and to know its main properties.

General characterization

Code

11641

Credits

6.0

Responsible teacher

Ana Margarida Fernandes Ribeiro

Hours

Weekly - 4

Total - 56

Teaching language

Inglês

Prerequisites

Knowledge on classical differential and integral calculus at a graduate level. Basic notions of toplogy and Functional  Analysis. Knowledge of the Lebesgue integral and of fundamental notions on Measure Theory.  

Bibliography

1.F. Friedlander and M. Joshi, Introduction to the Theory of distributions;

2.A. Kolmogorov and V. Fomin, Introductory real analysis;

3.W. Rudin, Functional Analysis;

4.R. Strichartz, A Guide to Distribution Theory and Fourier Transforms;

also:

5.L. Evans and M. Gariepy; Measure Theory and Fine Properties of Functions;

5.L. Schwartz, Théorie des Distributions;

6.L. Schwartz, Méthodes Mathématiques pour les Sciences Physiques.

7.Yosida, Functional Analysis.

Teaching method

Theorical/Problem solving sessions complemented by discussion sessions.

Evaluation method

1. Continuous evaluation

The classification in continuous evaluation is a score between 0 and 20 values and results from the classification obtained in two evaluation elements in the theoretical-practical evaluation component, with a weight of 80% (16 values) and a summative evaluation element with a weight of 20% (4 values). The two elements of theoretical-practical evaluation consist of two tests carried out in person with a duration of 2 hours. The element of summation is the study and presentation of a subject within the scope of the curriculum program.

Requirements for performing the tests: To perform any of the tests provided, students must enroll in Clip up to one week before the date of the test. In addition, at the time of the test, students must be carrying an examination notebook (blank) and an identification document in addition to complying with all the general rules of the faculty.

2. Exam

The exam is in person and lasts for 3 hours. All students who have not yet passed the course can take the exam. To take the exam, students must enroll in Clip up to one week before the exam date. In addition, at the time of the test, students must be carrying an examination notebook (blank) and an identification document in addition to complying with all the general rules of the faculty.

The final score is obtained by the weighted average of the test score and the sumative evaluation component with the weights indicated in the previous number.

3. Grade improvement exam

All students who wish to obtain grade improvement must comply with, for this purpose, the legal formalities of registration. In addition, the student must present himself to the exam, following the requirements indicated in point 2 above. Also, the final result is obtained as described above, in point 2.

Subject matter

1. Historical Context.

Differentiation of nonregular functions.

2. Preliminaries.

Frechet''s Spaces; Spaces C^k and C^{\infty}, Inductive topological limits and test functions.

3. Space of Distributions.

Linear Continuous forms; Locally Integrable Functions.

4. Convergence of sequences of distributions; Weak and Weak * topology; Caracterisation of the Convergence of

Distributions and fundamental properties.

5. Differentiation of distributions.

Differential operators; Fundamental Solutions; Convolution of distributions; General properties of convolution;

6. Tempered Distributions.

Smooth rapid descreasing functions; L^P spaces as tempered distributions; Properties of the Fourier Transform.

7. Application: Sobolev Spaces.

Programs

Programs where the course is taught: