# Theory of Distributions

## 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

10850

6.0

##### Responsible teacher

Ana Margarida Fernandes Ribeiro

Weekly - 3

Total - 56

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

There are two paper mid-term tests (50%+50%). Otherwise the student must pass the final exam. More detailed rules are available in the portuguese version.

## 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: