MATH5515 is a Mathematics Level V course. See the course overview below.

Units of credit: 6

Prerequisites:

Cycle of offering: Variable

Graduate attributes: the course will enhance your research, inquiry and analytical thinking abilities.

More information: this recent course handout (pdf) contains information about course objectives, assessment, course materials and the syllabus. (This pdf will usually be updated by the end of the first week of the semester.)

The Online Handbook entry contains up-to-date timetabling information.

If you are currently enrolled in MATH5515, you can log into the My eLearning Vista instance of this course.

This course develops the essentials of Fourier analysis and applications. Originated by Fourier for the study of heat, it helps us understand many aspects of signal processing, data compression, wavelets and differential equations, with important applications to physics, engineering and other areas of mathematics, including statistics.

The course begins with a review of the essential linear algebra, and then introduces Fourier's theory along historical lines. We work on the circle and then the line, establishing the general theory of (complex) Fourier series and then Fourier transforms.

We derive finite versions, including the Fast Fourier Transform (FFT), using commutative group theory, which will be introduced as needed, and give some applications to number theory. We show how the theory extends to more general commutative groups, and discuss the Poyntriagin duality theory.

Generalizations to simple hypergroups, and in particular to Bessel functions and some orthogonal polynomials will be discussed, and we sketch applications to the theory of distributions and differential equations, including the Sturm Liouville equation.

We may also discuss applications to the theory of distributions, and discuss appplications to statistics, for example to random walks and the Central Limit theorem.
In the last part of the course we introduce the theory of wavelets and give some applications.