ADVANCED ANALYSIS 1

ADVANCED ANALYSIS 1

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Code
61683
ACADEMIC YEAR
2021/2022
CREDITS
8 credits during the 1st year of 9011 Mathematics (LM-40) GENOVA
SCIENTIFIC DISCIPLINARY SECTOR
MAT/05
LANGUAGE
Italian
TEACHING LOCATION
GENOVA (Mathematics)
semester
2° Semester
Teaching materials

OVERVIEW

Language: Italian

AIMS AND CONTENT

LEARNING OUTCOMES

The course aims to provide an introduction to distribution theory and some of its applications

Teaching methods

Teaching style: In presence

SYLLABUS/CONTENT

Convolution of two functions.

Locally convex topological vector spaces.

Definition of a distribution. The space of test functions and the space of distributions. Order of a distribution. Distributions of order 0. Calculus on distributions. Compactly supported distributions. Distributions supported in a point. Regularization of divergent integrals. Convolution of a distribution with a test function. Convolution of two distributions. Analytic continuation of distributions.

Fundamental solutions of constant coefficients operators. Hypoelliptic operators. Parametrics. The local structure of distributions.

Fourier transform. Temperated distributions. Fundamental solutions of Laplace and heat operators. The Cauchy problem for the heat operator. Fourier transform of compactly supported distributions. Paley-Wiener theorems. Sobolev spaces (definition). The Cauchy problem for the wave operator. Finite speed of propagation. Fundamental solution of the wave operator.

The Schwartz kernel theorem.

TEACHERS AND EXAM BOARD

LESSONS

Teaching methods

Teaching style: In presence

LESSONS START

The class will start according to the academic calendar.

EXAMS

Exam description

Oral.