Math 262 / CME 372: Handouts





Course handouts:

  • Course Organization .html.
  • Course Survey: .pdf.


  • Lecture 1: Time-invariant (linear) operators, convolutions, continuous-time Fourier transform
  • Lecture 2: Fourier inversion formula, convolution theorem, central limit theorem via Fourier transforms
  • Lecture 3: Parseval-Plancherel theorem, Fourier transform of square-integrable functions, Fourier transform of distributions, Fourier transform in higher dimensions
  • Lecture 4: The (Weyl-) Heisenberg uncertainty principle and its interpretation in quantum mechanics
  • Lecture 5: Poisson summation formula, sampling and the aliasing formula, Shannon's sampling theorem
  • Lecture 6: Discrete convolutions, Fourier series, another look at Shannon's sampling theorem
  • Lecture 7: Numerical accuracy of the trapezoidal rule, Fourier transform of finite signals, FFTs
  • Lecture 8: X-ray tomography, X-ray propagation and Beer's law
  • Lecture 9: X-ray tomography, Radon transform, backprojection, Radon inversion formula
  • Lecture 10: X-ray tomography, ill-posedness of the inverse problem, regularized inversion
  • Lecture 11: Non-uniform fast Fourier transforms (NUFFTs)
  • Lecture 12: Magnetic Resonance Imaging (MRI), nuclear magnetic resonance (NMR), Bloch phenomenological equations, relaxation times
  • Lecture 13: Magnetic Resonance Imaging (MRI), simple imaging experiment, signal equation
  • Lecture 14: Magnetic Resonance Imaging (MRI), roles of relaxation times, selective excitation Lecture 15: Wave optics, the phenomenon of diffraction, history, Fresnel-Kirchhoff integral formula
  • Lecture 16: Rayleigh-Sommerfeld diffraction theory, Fresnel diffraction, Fraunhoffer diffraction, examples
  • Lecture 17: Lenses, thin lenses, Fourier transform properties of thin lenses
  • Lecture 18: Image formation, relation between object and image, effects of diffraction
  • Lectures 19: The Wiener filter: Karhunen-Loeve decomposition of stochastic processes, stationary processes, estimation of Gaussian processes