35005 Lebesgue Integration and Fourier Analysis
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Credit points: 6 cp
Result type: Grade and marks
Requisite(s): (35007 Real Analysis AND (33230 Mathematics 2 OR 33290 Statistics and Mathematics for Science OR 37132 Introduction to Mathematical Analysis and Modelling))
These requisites may not apply to students in certain courses. See access conditions.
Description
Fourier Analysis shows how to decompose a function (or a signal) as a sum (or integral) of sines and cosines. It is used in many applications including Engineering, Finance, Biology; in fact, in any system where periodic phenomena are important. This subject introduces a more sophisticated version of integration, developed by Henri Lebesgue, to handle the demands of Fourier Analysis. The technique is required for many current applications of mathematics, and this subject will discuss a number of those applications.
Subject learning objectives (SLOs)
Upon successful completion of this subject students should be able to:
| 1. | Build one's mathematical language and arguments to explain definitions and theorems relevant to the subject, presenting the information with the highest standards of professional rigour |
|---|---|
| 2. | Demonstrate how language is used when referencing definitions and theorems to one's peers |
| 3. | Analyse and synthesise the methods of proofs of the theorems and be able to prove these theorems. Validate and test ideas in the theory of the integral and Fourier Analysis. |
| 4. | Hypothesise and prove or disprove results the students have not seen before, using methods of mathematical analysis |
| 5. | Apply results from the subject to problems arising in related disciplines including probability theory and engineering. |
Course intended learning outcomes (CILOs)
This subject also contributes specifically to the development of following course intended learning outcomes:
- Demonstrate theoretical and technical knowledge of mathematical sciences including calculus, discrete mathematics, linear algebra, probability, statistics and quantitative management. (1.1)
- Evaluate mathematical and statistical approaches to problem solving, analysis, application, and critical thinking to make mathematical arguments, and conduct experiments based on analytical, numerical, statistical, algorithms to solve new problems. (2.1)
- Design creative solutions to contemporary mathematical sciences-related issues by incorporating innovative methods, reflective practices and self-directed learning. (4.1)
- Use succinct and accurate presentation of reasoning and conclusions to communicate mathematical solutions, and their implications, to a variety of audiences, using a variety of approaches. (5.1)
Contribution to the development of graduate attributes
This subject also contributes specifically to the development of the following Graduate Attributes to prepare students
for professional practice:
Graduate Attribute 1 – Disciplinary knowledge
Activities in this subject develop practical skills to analyse problems which involve advanced mathematical analysis. Both the formal mathematical framework of measure theory and many of its common applications in diverse applied fields are explored.
Graduate Attribute 2 - Research. Inquiry and Critical Thinking
The subject will develop students' capabilities in quantitative thinking and introduce them to research themes and results in mathematical analysis which have had, and continue to have, significant impacts on mathematics.
Graduate Attribute 4 – Reflection, Innovation, Creativity
The assignments focus on the understanding of mathematical systems and their analysis in both formal mathematical language and in peer-to-peer (non-mathematical) language. Students are encouraged to formulate new ideas and to test their validity.
Graduate Attribute 5 – Communication
The assignments require students to write in clear English explaining their mathematical reasoning and its
conclusions. Ability to communicate the ideas of the subject's development and their interrelations is crucial.
All assessment tasks require appropriate presentation of information, reasoning and conclusions and require students
to gain meaning from instructions (written or verbal) and problem statements.
Teaching and learning strategies
Workshops: interactive learning based on the topic list. Students will undertake pre-reading of the material to be discussed (available on Canvas), attend a discussion with the lecturer and other students to clarify their understanding of the topics and answer questions in class. Students will receive verbal feedback in class and written feedback on their assignments.
Tutorials: student led exercises. Students will propose solutions to the tutorial exercises and discuss their solutions with the teacher and other class members. Students will be encouraged to work collaboratively on the tutorial exercises, both inside and outside allocated class time.
Content (topics)
1. Review of basic analysis and Riemann Integration
2. Lebesgue measure and measurable sets
3. Convergence of measurable functions: Littlewood's three principles
4. The Lebesgue integral: the big theorems of Lebesgue integration
5. L^p spaces and other Banach spaces
6. Fourier series and Fourier transforms
7. Convergence of Fourier series: the L^2 theory
8. Fourier transforms and applications to solutions of differential equations
Assessment
Assessment task 1: Assignment 1
| Intent: | This assessment task contributes to the development of the following graduate attributes: 1. Disciplinary Knowledge 2. Research, Inquiry and Critical Thinking 4. Reflection, Innovation and Creativity 5. Communication |
|---|---|
| Objective(s): | This assessment task addresses subject learning objective(s): 1, 2, 3 and 4 This assessment task contributes to the development of course intended learning outcome(s): 1.1, 2.1, 4.1 and 5.1 |
| Type: | Exercises |
| Groupwork: | Group, individually assessed |
| Weight: | 25% |
| Criteria: | Capacity in calculations and facility in constructing valid mathematical arguments. |
Assessment task 2: Assignment 2
| Intent: | This assessment task contributes to the development of the following graduate attributes: 1. Disciplinary Knowledge 2. Research, Inquiry and Critical Thinking 4. Reflection, Innovation and Creativity 5. Communication |
|---|---|
| Objective(s): | This assessment task addresses subject learning objective(s): 1, 2, 3, 4 and 5 This assessment task contributes to the development of course intended learning outcome(s): 1.1, 2.1, 4.1 and 5.1 |
| Type: | Exercises |
| Groupwork: | Group, individually assessed |
| Weight: | 25% |
| Criteria: | Capacity in calculations and facility in constructing valid mathematical arguments. |
Assessment task 3: Assignment 3
| Intent: | This assessment task contributes to the development of the following graduate attributes: 1. Disciplinary Knowledge 2. Research, Inquiry and Critical Thinking 4. Reflection, Innovation and Creativity 5. Communication |
|---|---|
| Objective(s): | This assessment task addresses subject learning objective(s): 1, 2, 3, 4 and 5 This assessment task contributes to the development of course intended learning outcome(s): 1.1, 2.1, 4.1 and 5.1 |
| Type: | Exercises |
| Groupwork: | Group, individually assessed |
| Weight: | 50% |
| Criteria: | Capacity in calculations and facility in constructing valid mathematical arguments. |
Minimum requirements
Students will be expected to achieve a total mark of 50% or greater to pass this subject