Gerard Sleijpen,

Lecture Notes on Fourier theory (Nov., 2011, ed., 2014) (.ps.gz, .pdf)

Transparancies and handouts of the transparancies (in pdf)

• Lecture 1: “Introduction Fourier theory, wavelets”, transp., handouts.
• Lecture 2: “Fourier Series”, transp., handouts.
• Lecture 3: “Fourier Integrals”, transp., handouts.
• Lecture 4: Computer session
• Lecture 5: “Fourier Transform in Practice: examples”, transp., handouts.
• Lecture 6: “Computational aspects of Fourier Transforms”, transp., handouts.
• Lecture 7: “Convolution products”, transp., handouts.
• Lecture 8: Computer session
• Lecture 9: “Filtering”, transp., handouts.

Additional resources:

• Real and Complex Analysis by Walter Rudin, 3rd ed., McGraw-Hill International Edition.
  
• Chapter 6 of A First Course in Computational Physics by Paul L. DeVries, John Wiley & Sons, 1994.
• Principles of Computerized Tomographic Imaging, by Avinash C. Kak and Malcolm Slaney.
  

Some Useful Links

• Ooura's FFT package
• KUIM Image Processing System
• http://www.colorado.edu/physics/2000/tomography/

Applications

• Computerized Tomography
• MRI (Magnetic Resonance Imaging):
  • The Basics of MRI, (text as well as annimations) by Joseph P. Hornak
  • Interactive course about MRI physics and mathematics (e-MRI), by Denis Hoa, Antoine Micheau, Gérald Gahide.
  • Annimations to demonstrate the physical principles of MRI
  • In Wikipedia: Nuclear magnetic resonance and MRI
  • From quantum mechanics to Fourier transform:
    • In Physical Principles and Clinical Applications of NMR, R A Lerski (ed): "An introduction" (pdf), by Vennart, W.
    • Chapter 2 (pdf) of Principles of MRI, Thesis, Nottingham, by Stuart Clare
• Crystallography:
  • Kevin Cowtan's Book of Fourier
  • Interactive Tutorial about Diffraction     (Department of Crystallography in Würzburg)
• Fraunhofer diffraction
• MP3 and other compression techniques for audio signals
  • Psychoacoustic model
  • Scott N. Levine's page (with a link to his Ph.D. thesis)
  • Kim Jong-Wha thesis (discusses also a wavelet based approach)
  • Hossein Najafzadeh-Azghandi thesis, J. van der Vorm M.Sc. thesis
• Image compression (JPEG)
  • Course (Simon Fraser University, CS)
• Partial Differential Equations
• Stablity analysis of numerical methods for PDEs
• Radar
• Speech analysis
• ...

Applets

• Hearing Fourier series
• Diffraction with two (point) slits.
• Here you can find more applets on related topics.