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# Introduction to the Theory of Distributions and Applications

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## Overview

The aim of the course is to make the interested student knowledgeable in the basic notions of the Theory of Distributions and its use in concrete applications. Typically, a person working in the field of Electronic Engineering, will be able to apply the tools learned in the course, to address problems coming from Signal Theory. The final chapter of the course can also be seen as an introduction to Partial Differential Equations, paving therefore the way to further mathematical studies for interested people. Finally, the Theory of Distributions is a beautiful piece of Mathematics, and the course is surely a good opportunity also for all those persons who are simply interested in broadening their mathematical knowledge, without an immediate practical aim.

## Syllabus

First Chapter: A primer about distributions

Unit 1 - Introduction; Definition of the space D(Ω); Definition of distributions Unit 2 - The L^1_{loc} space and the notion of convergence Unit 3 - Derivatives in the sense of distributions; Convergence in the sense of distributions Unit 4 - Completeness of the space of distributions; Simple examples of distributions

Second Chapter: Main operations with distributions

Unit 1 - Multiplication of a distribution by a C^∞ function; Leibniz’s Formula for the product Unit 2 - Composition; Restriction; Tensor Product Unit 3 - The Fundamental Theorem of Calculus in the context of distributions Unit 4 - Support of a distribution; Compactly supported distributions; Extension from D to C^∞ Unit 5 - Division in the sense of distributions

Third Chapter: Fourier Transforms for functions in L^1 and L^2

Unit 1 - The Fourier Transform in L^1 Unit 2 - Inversion of the Fourier Transform in L^1 Unit 3 - The Fourier Transform in L^2 Unit 4 - Inversion of the Fourier Transform in L^2; Unitary operators

Fourth Chapter: Tempered distributions

Unit 1 - Introduction to the space of tempered distributions Unit 2 - The Fourier Transform for tempered distributions Unit 3 - Simple applications of the Fourier Transform for tempered distributions

Fifth Chapter: Convolution

Unit 1 - Convolution between a D function and a D’ distribution Unit 2 - Further examples of convolution between a function and a distribution; Convolution between two proper distributions Unit 3 - The Theorem of Convolution for the Fourier Transform of tempered distributions Unit 4 - The Paley-Wiener Theorem Unit 5 - Simple applications of the Paley-Wiener Theorem

Sixth Chapter: Applications to Linear Partial Differential Equations

Unit 1 - The fundamental solution of the Laplace equation; An application Unit 2 - The fundamental solution of the heat equation; An application Unit 3 - The fundamental solution of the wave equation; An application

Ugo Gianazza

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