Dr. Gilbert Strang
Massachusetts Institute of Technology

Tuesday, January 23, 2001
Center for Environmental Sciences & Technology Management (CESTM) Auditorium
University at Albany
4 p.m. to 6 p.m.

Refreshments Served
Sponsored by the University at Albany�s Division for Research

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Three Matrices and Their Applications

First, ideas in the engineering literature are discussed on an * integer * form of the Fast Fourier Transform. The goal is to have a close approximation that can be quickly executed with low power (and without floating point multiplications). The IntFFT starts with a matrix factorization.

Next, the localized eigenvectors that appear when a few entries are changed in a familiar tridiagonal Toeplitz matrix are discussed. The eigenvalues of the modified matrix allow a surprisingly simple formula.

So much of mathematics is involved with the representation of functions. A central example in pure and applied mathematics is the Fourier series. Its discrete version is computed by the Fast Fourier Transform, which is the most important algorithm of the last century. The Fourier basis is terrific � but imperfect. The basis functions are global instead of local, and they give poor approximation at a discontinuity (Gibbs phenomenon). New functions are being developed for interpolation and approximation and compression and many other applications. * An outstanding problem is video compression for the Internet. *

Four properties that are desired are: local basis, easily refined, fast to compute, and good approximation by a few terms. Splines and finite elements achieve the first three, but they do not allow compression; if terms are removed in a spline expansion this leaves blank intervals. So wavelet construction to permit compression of data are used � which is needed in so many modern applications where the volume of data is overwhelming. Wavelets have two types of basis functions, one for averages and the other (the wavelets themselves) for details at all scales. When the details are not necessary they can be compressed away to leave a smoothed signal. That construction has entered the new IEEE standards for signal processing. An explanation will be given on how the construction is achieved with two filters (where one filter would fail). In matrix terms, we get a banded block Toeplitz matrix with a banded inverse.

For directions to CESTM go to www.albany.edu. For additional information, call (518) 442-3332.

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