Professor Anthony Dooley, University of Technology Sydney
Associate Professor Po-Lam Yung, Australian National University
The Fourier transform is an ubiquitous tool that allows one to represent a function as superposition of different frequencies. Its analysis led to substantial progress in the study of partial differential equations and analytic number theory, and has found important applications in diverse fields such as information theory and physics.
One early connection came from the Hilbert Transform:
This is of great interest from many points of view, from complex analysis to boundary value problems for the Laplacian and signal processing (this last connection comes from the formula
a mechanism with which one can can switch on/off a signal). In 1928, Riesz proved that H is bounded onLp(R) for all 1 < p < ∞. This is the beginning of a rich and fruitful theory of singular integrals,concerning operators whose kernels (such as 1/y) barely fails to be integrable on the domain (R in this case). It provides the technical tool required for obtaining a Fourier representation of a general Lp function on R.
Participation in all lectures and tutorials is expected.
For those completing the subject for their own knowledge/interest, evidence of at least 80% attendance at lectures and tutorials is required to receive a certificate of attendance.
You are expected to have taken a course on elementary analysis, at the level of W. Rudin’s “Principles of Mathematical Analysis”, and it will be helpful for you to have read:
Take this QUIZ to self-evaluate and get a measure of the key foundational knowledge required.