Introduction to the Theory of Distributions and Applications

Introduction to the Theory of Distributions and Applications

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Although rigorous in the discussion of the different topics, the course will mainly focus on the applicable side of the results it will present, with a special care for problems from Physics and Engineering. Therefore, there will not be an emphasis on proofs (with all the possible technical complications), but rather on how the different notions are related to one another, and how they can be successfully employed to tackle some interesting issues coming from other branches of science.

Who is this course for?

This is basically a course at the Master Level. Even though the hardest mathematical aspects of the theory will not be dealt with here (for example, for anyone even barely familiar with them, there will be no mention of topologival vector spaces), nevertheless a certain level of mathematical knowledge is a fudamental prerequisite, which cannot be done away with.

What do I need to know?

A person interested in this course can take it successfully, provided he/she has already basic proficiency (for example at a bachelor degree level) in Linear Algebra, One-variable Calculus, Multivariable Calculus, Metric and Normed Spaces and their introductory feature.

What will I learn?

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.

Course Structure

As clearly explained above, the course is quite technical, and a satisfactory description of the contents of each unit would be too long. It suffices to say that what we are trying to convey is more or less the following:

First chapter: what is all about
Second chapter: classical operations can be defined for distributions too, but they require an extra and unexpected care.
Third chapter: this is a detour, namely a crash course on "classical Fourier transform", which acts ans an introduction to the next chapter.
Fourth chapter: we give a look at the notion of Fourier transform in the context of distributions.
Fifth chapter: this is something that is quite interesting and important, mainly (but not exclusively) for people interested in Signal Theory.
Sixth chapter: few (very few!) words about the wonderful world of Partial Differential Equations, and how they can be dealt with using distributions.

Here are more technical details about the contents of all the different chapters.

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

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