Real and harmonic analysis

lmat1322  2024-2025  Louvain-la-Neuve

Real and harmonic analysis
5.00 credits
30.0 h + 30.0 h
Q2
> Schedule
Teacher(s)
Ponce Augusto;
Language
French
> English-friendly
Prerequisites
It is recommended that the student be familiar with the basic concepts of real analysis as developed in LMAT1122 and be familiar with or in the process of becoming familiar with notions of integration in Euclidean spaces as developed in LMAT1221.
Some familiarity with the language of functional analysis as developed in LMAT1321 may be helpful, but is not essential.
Main themes
The course covers the basics of measurement theory and Fourier analysis.
Learning outcomes

At the end of this learning unit, the student is able to :

1 At the end of this activity, students will be able to :
  • define mathematically the fundamental objects of the course,
  • state and prove the course's propositions and theorems,
  • illustrate definitions, propositions and theorems with examples, counter-examples and applications,
  • apply demonstration methods learned in the course to similar situations.
Students will have progressed in their ability to :
  • identify the unifying aspects of different situations and experiences,
  • reason within the framework of the axiomatic method,
  • construct and write a demonstration independently, clearly and rigorously.
 
Content
The course will cover elements of real analysis and harmonic analysis in Euclidean space:
  • Kurzweil-Henstock integral,
  • Fundamental Theorem of Calculus,
  • Lebesgue's differentiation theorem,
  • convolution product,
  • Fourier transform.
Teaching methods
The learning activities consist of lectures and practical sessions.
The lectures aim to introduce the fundamental concepts, to motivate them by showing examples and establishing results, to show their reciprocal links and their links with other courses in the Bachelor of Mathematical Sciences program.
The practical sessions aim at deepening the concepts discussed in the lecture.
Evaluation methods
The assessment will take the form of a continuous assessment, based on mandatory assignments to be submitted throughout the term. Participation in lectures is mandatory. In the event of a second registration for the exam, the assessment will take the form of a written exam covering the entire subject.
Online resources
Additional documents on Moodle.
Bibliography
Le cours sera basé sur des extraits des références suivantes :
  • A. Ponce et J. Van Schaftingen. LMAT1121 - Analyse mathématique 1, DUC, Louvain-la-Neuve, 2016
  • A. Ponce et J. Van Schaftingen. LMAT1221 - Analyse mathématique 3, DUC, Louvain-la-Neuve, 2016
  • A. Ponce. Elliptic PDEs, measures and capacities, EMS Tracts Math. 23, European Mathematical Society (EMS), Zürich, 2016
  • P. Mironescu. Mesure et intégration. Polycopié parcours L3 math, Université Claude Bernard, Lyon, 2020
Faculty or entity
> MATH


Programmes / formations proposant cette unité d'enseignement (UE)

Title of the programme
Sigle
Credits
Prerequisites
Learning outcomes
Additionnal module in Mathematics
APPMATH
5
Bachelor in Mathematics
MATH1BA
5