Real and harmonic analysis

lmat1322  2023-2024  Louvain-la-Neuve

Real and harmonic analysis
5.00 credits
30.0 h + 30.0 h
Q2
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 the abstract theory of measure and harmonic anaylsis elements in Euclidean space :
  •     Fréchet measure and integral,
  •     decompositions of measures,
  •     integral convergence theorems,
  •     Lebesgue differentiation theorem,
  •     product measure and theorems of Fubini and Tonelli,
  •     change of variables theorem,
  •     convolution product,
  •     series and 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
Skill acquisition will be assessed in a final exam.
Questions will require :
  •     render material, including definitions, theorems, proofs, examples,
  •     select and apply methods from the course to solve problems and exercises
  •     adapt methods of demonstration from the course to new situations,
  •     synthesize and compare objects and concepts.
Assessment will include :
  •     the knowledge, understanding and application of the various mathematical objects and methods of the course,
  •     the rigor of the developments, proofs and justifications,
  •     the quality of the writing of the answers.
Online resources
Additional documents on Moodle.
Teaching materials
  • R. G. Bartle, The Elements of Integration and Lebesgue Measure, Wiley, 1966. ISBN-10 : 0471042226
  • 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