# Introduction to the Theory of Distributions and Applications

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

ENROLL IN COURSE LINK: iversity.org/en/courses/introduc...tions

## Lessons hide description |
---|

####
**Math 1A/1B. Pre-Calculus: Algebra and Geometry Review - UCI**
51 lessons

This open course provides the student with review of intermediate algebra, geometry, and trigonometry. It is designed for those want to be successful in an introductory college calculus course, but wo

####
**Physics 50: Math Methods - UCI**
75 lessons

Description: Mathematica and its applications to linear algebra, differential equations, and complex functions. Fourier series and Fourier transforms. Other topics in integral transforms. Filmed Sprin

####
**Earth System Science 5: The Atmosphere - UCI**
19 lessons

Description: The composition and circulation of the atmosphere with a focus on explaining the fundamentals of weather and climate. Topics include solar and terrestrial radiation, clouds, and weather p

####
**Mathematical Biology - UCI**
27 lessons

Introduction to Mathematical Modeling in Biology is taught by Professor German A. Enciso, Ph.D. The book for this class is "Mathematical Models in Biology" by Leah Edelstein-Keshet, SIAM, 2005 and co

####
**The Power of Macroeconomics - UCI**
69 lessons

Description: In this course, you will learn all of the major principles of macroeconomics normally taught in a quarter or semester course to college undergraduates or MBA students. Perhaps more im

####
**The Power of Microeconomics - UCI**
61 lessons

Description: In this course, you will learn all of the major principles of microeconomics normally taught in a quarter or semester course to college undergraduates or MBA students. Perhaps more import

####
**Einstein's General Relativity and Gravitation - UCI**
24 lessons

The graduate General Relativity course is taught by Professor Herbert W. Hamber, Ph.D. Description: "Einstein's General Relativity and Gravitation" is taught at UCI as Physics 255. Topics covered: A

####
**Intro to Statistics**
960 lessons

Statistics is about extracting meaning from data. In this class, we will introduce techniques for visualizing relationships in data and systematic techniques for understanding the relationships using

####
**Introduction to Probability and Statistics B - UCI**
15 lessons

Description: UCI Math 131B is an introductory course covering basic principles of probability and statistical inference. Point estimation, interval estimating, and testing hypotheses, Bayesian approac

####
**Introduction to Probability and Statistics A - UCI**
16 lessons

Description: UCI Math 131A is an introductory course covering basic principles of probability and statistical inference. Axiomatic definition of probability, random variables, probability distribution

**TAG Cloud**

**Web Stats**

Active courses: 1,180

Lessons: 27,706

Data: 83 GB

Online: 107 users

News about new courses