Course PM
MMA410 - Fourier and Wavelet Analysis
course diary

Here I will try to keep an updated account of what has been done so far in the course.
date what we did comments
29/10 - 1/11
  • We have mostly worked on Ch. 1 in Bergh: Notes on generalized functions.
  • definition and some properties of the Fourier transform
  • the inversion formula
  • the class S
  • convolution and autocorrelation
  • the tempered distributions, S' , differentiation, multiplication by functions, the Fourier transform.
  • some examples: Dirac measure, the finite part of x-3/2 H(x)
5/11 - 8/11
  • We have finished Chapter 1 in Bergh
  • Most of the relevant details in Bracewell: ch1 -ch10. Some more about power spectra, and some inequalities related to Fourier analysis remain.
  • Rules for computing DFT
  • The fast Fourier transform
  • Cost of computing the FFT
  • The uncertainty relation
  • Fourier transforms and tempered distributions in higher dimension.
  • tensor products of tempered distributions
  • Tensor products of tempered distributions
  • Some formulas related to mulitdimensional fourier transforms
  • The Hankel transform (in 2 and more dimensions)
  • A little about the assignement 1: How to experiment with filters and audio signals from e.g. music.
  • The Abel transform
  • The Radon transform: its definition, and the inversion theorem (without proof so far)
21-23 /11
  • note that there only lectures this week.
  • The Radon transform: proof of the inversion formula
  • The Hilbert transform and the analytical signal
  • The fourier transform of homogeneous distributions, and in particular of (-z)-1/2
  • The Uncertainty principle
  • A little about probability theory: a proof of the central limit theorem
  • Tempered measures, positive measures and positive definite distributions.
  • Beginning of discrete filter theory: linear time invariant and causal filters, FIR and IIR ...

Bernt Wennberg <>