00020 - Advanced Analysis

Learning outcomes

 Working knowledge of modern distribution theory and Fourier transform as the basic tools for the modern theory of Partial Differential Equations.

Course contents

 Summary:

The method of characteristics for the solution of partial differential equations of order one. Theory of distributions and of their Fourier transform.

Detailed description:

Method of characteristics, I: introduction; motivation.

Method of characteristics, II: solution of systems of PDEs. Topology of C^k(D).

Test functions and their properties. Construction of test functions.

Definition of distribution D'(D); examples of distributions.

Poisson equation ``in the sense of distributions". Regular distributions; singular distributions. A characterization of D'(D).

TLS topology: definition and main properties.

Repesentation of D'(D) as the dual space of test functions in the TLS topology.

Localization and welding of distributions. 

Motivation: support of continuous functions. Support of a distribution. Distributional derivative; examples.

Representation of D'(D) as derivatoves of essentially bounded functions.

Pointwise product of a distribution by a smooth function: definition; examples and properties.

Regularity of a function singular at a point of an interval of the real whose distributional derivative is in L^1 of said interval.

Extension of the domain of a distribution: distributions with compact support. Characterization as the dual space of the smooth functions in the metric topology induced by the seminorms of C^\infty(D).

Schwartz's example.

Representation of the kernel of a distribution with compact support.

Decomposition of a distribution supported on a singleton as a linear combination of Dirac's delta and its derivatives.

Convolution of a distribution with a test function: definition and properties.

Cnvolution of a distribution with compact support with a smooth function: definition and properties.

Regularization of a distribution; density of the test functions in D'(D). Translation operator.

Convolution of two distributions with at least one having compact support: definition and properties (proof of commutativity).

Convolution of three distributions with at least two having compact support: definition and properties (proof of associativity).

Fourier transform on L^1(R^n): definition and main properties.

The Schwartz class, S: definition and main properties (among these, immersion in L^1(R^n)).

The Fourier transform is an isomorphism on S. 

S is (continuously) embedded in L^2(R^n). Parseval's formula.

Tempered distributions, S’: motivation; definition; continuous embedding in L^p(R^n).

Fourier transform on S’: definition; isomorfism properties; inversion formula on S’.

Fourier transform on L^2(R^n); Plancherel theoem.

Paley-Wiener-Schwartz theorem.

 

 

 

 

Readings/Bibliography

1. L. Hörmander: Linear Partial Differential Operators, Springer Edizione del 1969, Chapter 1.
2. L. Grafakos: Classical Fourier analysis, 3d edition (2014), Chapter 2, sections 2.2; 2.3; 2.4; Chapter 5, sections sezione 5.1.1.
3. L. Hörmander: Linear Partial Differential Operators I, Springer 2nd edition 1990, chapters 2, 3, 4, 6, 7.
   
4. F.G. Friedlander. Introduction to the theory of distributions. Second edition. With additional material by M. Joshi. Cambridge University Press, Cambridge, 1998.
   
5. C. Parenti e A. Parmeggiani, Algebra Lineare ed Equazioni Differenziali Ordinarie, 2nd edition, Springer Unitext 117, capitolo 5.
6. C. Zuily: Eléments de distributions et d'équations aux dérivées partielles. Dunod.

Teaching methods

In-person lectures consisting of theory, examples, exercises and applications, also aimed to the applied math curriculum.

Assessment methods

Course evaluation is based on a final, oral examination consisting of an exposition of a topic chosen by the student on the material covered in class, to be followed by questions pertaining to theorems proved in class, and/or the solution of exercises assigned during the course.

 

This course is on of the two-modules integrated course on Advanced Analysis and Differential Geometry: the course grade is the average of the grades earned in the two modules. A grade of 29 in the analysis module does not preclude from earning 30 Cum Laude for the integrated course

Office hours

See the website of Loredana Lanzani