Fourier Analysis and Lebesgue Integration
|Unit level:||Level 6|
|Teaching period(s):||Semester 2|
|Offered by||School of Mathematics|
|Available as a free choice unit?:||N
Students are not permitted to take MATH41012 and MATH61012 for credit in an undergraduate programme and then a postgraduate programme.
To give an introduction to Lebesgue's theory of measure and integration on the set of real numbers R. To use this to find an appropriate setting in which to understand the convergence of Fourier series.
It is often convenient to represent functions as Fourier series. However, the convergence of such series is a delicate issue closely related to the theory of integration. A standard approach to integration on the real line, formalised by Riemann, is based on partitioning the domain into smaller intervals. This approach works in many situations but there are simple examples for which it fails. In the early 1900s, H. Lebesgue produced a better theory in which the key idea is to extend the notion of length from intervals to more complicated subsets of R. This started an area of mathematics it its own right, called Measure Theory. Most generally, this is about how one may sensibly assign a size to members of a collection of sets. One application of Lebesgue's ideas is that one can introduce a vector space of functions in which Fourier series appear in a natural way.
This course will appeal to students who have enjoyed MATH20101 or MATH20111 and MATH20122. It will be useful to student taking probability courses in years three and four since the ideas of measure theory have a central role in probability theory.
On successful completion of this course unit students will be able to:
- formulate the Fourier series associated to a periodic function;
- describe what is meant by a function being simple, measurable, integrable, and verify if certain functions satisfy such properties;
- define a σ-algebra, Lebesgue measure on R and construct integrals using measures;
- prove the convergence theorems for the Lebesgue integral and use them to calculate the Lebesgue integral for certain infinite series;
- prove that the set of square integrable periodic functions forms a Hilbert space;
- relate the Fourier series of a periodic function to the Hilbert space of square integrable periodic functions;
- Other - 20%
- Written exam - 80%
Assessment Further Information
- Mid-semester coursework: weighting 20%
- End of semester examination: three hours weighting 80%
- Fourier series, convergence and Dirichlet's Theorem. [3 lectures]
- Revision of countable and uncountable sets, the Cantor set. [2 lectures]
- Riemann's approach to integration. Lesbesgue measure on R, Borel sets, measurable sets and functions, construction of the Lebesgue integral. [8 lectures]
- Limit theorems for the Lebesgue integral. [3 lectures]
- Square integrable functions and Fourier series, Hilbert spaces. [6 lectures]
The lectures will be enhanced by additional reading on the costruction of Lebesgue measure, existence of non-measurable sets, and general measures. Reading material will be provided.
- J. Franks, A (Terse) Introduction to Lebesgue Integration, American Mathematical Society, Student Mathematical Library, 2009.
- H. S. Bear, A Primer of Lebesgue Integration, Academic Press, 1995.
Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding. Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback. Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.
- Lectures - 22 hours
- Tutorials - 11 hours
- Independent study hours - 117 hours