# 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

##### Code

10850

##### Credits

6.0

##### Responsible teacher

Ana Margarida Fernandes Ribeiro

##### Hours

Weekly - 3

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

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.*