AMATH 391: From Fourier to Wavelets

Offered in the fall term in odd-numbered years.

Brief description:

Wavelets are setting the new standards for digitization, for example MPEG-4 for high-definition TV and JPEG-2000 for digital still images. This course covers the mathematical development from Fourier analysis to wavelets, with special emphasis on the conversion of a signal from the analog (continuous) to the digital (discrete) domain and its subsequent reconstruction. The material with linear algebra, is interspersed with concrete examples and numerical applications such as experimentation with audio signals.

Selected topics:

Inner product spaces, least squares approximations; Fourier series (FS), Fourier transform (FT); classical sampling theorem; decay and smoothness under FS and FT, uncertainty principle; denoising, oversampling, aliasing; Haar wavelet, multiresolution analysis; data compression by thresholding; singularity detection; spline wavelets, outlook on multidimensional digitization and image processing.

Prerequisites:

(AMATH 231 or SYDE 252 or ECE 342) and (MATH 136/146/115 or SYDE 114).

Intended audience:

The level of presentation of the material will make the course suitable for undergraduates in Applied Mathematics, Computational Mathematics, Pure Mathematics, Computer Science, Physics and Engineering.

Textbook:

A. Boggess and F. J. Narcowich, "A First Course in Wavelets with Fourier Analysis", Prentice Hall, Upper Saddle River, 2001.