SIAM AG on Orthogonal Polynomials and Special Functions


Extract from OP-SF NET

Topic #1 --------------- OP-SF NET --------------- September 12, 1996

SIAG/OS Events at 1996 SIAM Annual Meeting

From: Willard Miller, Jr.

SIAG/OS sponsored a minisymposium on Modern Topics in Orthogonal Systems on Tuesday July 23, at the July 22-26,1996 SIAM Annual Meeting in Kansas City, Missouri. Due to a cancellation by Robert Gustafson, only one major topic was covered: Wavelets. There were three talks.

Gilbert Walter organized the session and led off with "Improving wavelet approximations." Although discrete wavelets generally have superior convergence properties compared to classical orthogonal systems, they share the Gibbs' phenomenon shortcoming - which causes errors at the edge of a truncated signal or image. It was shown by Shim and Volkmer that this always happens for orthogonal approximations for all continuous wavelets with sufficient decay. Walter presented ways of avoiding this shortcoming for wavelets.

Peter R. Massopust spoke on "Multiwavelets, Multiresolution Schemes, and Hyperbolic Conservation Laws." He presented multiresolution schemes based on multiwavelets. These schemes employ a combination of interpolation and direct evaluation. He showed how such multiresolution schemes can be used to obtain accurate and computationally efficient numerical weak solutions of partial differential equations arising in computational fluid dynamics.

Truong Nguyen spoke on "Image Coding Using Shift-Invariant Dyadic Wavelet Transform", joint work with Y. Hui and C.W. Kok. He proposed a new class of wavelet filters, shift-invariant wavelet filters, for the purpose of image compression. The proposed shift-invariant wavelet transform has better shift-invariant properties compared with the conventional dyadic wavelet transform, without changing the structure. He proposed and evaluated two bit allocation schemes, suitable for the proposed shift-invariant wavelet transform coding, and presented experimental results showing that the shift-invariant wavelet transform has better energy compaction properties in image coding than the conventional wavelet transform.

Later in the morning Carl de Boor gave the John von Neumann Lecture on (multivariate) "Polynomial Interpolation." This was one of the highlights of the meeting.

Two of the SIAG-OS officers, Martin Muldoon and Willard Miller, participated in the week's events and held an impromptu business meeting to discuss possible minisymposium topics for the SIAG-OS at the 1997 SIAM Annual Meeting (July 14-18, Stanford University). The 1997 meeting themes most relevant to our group are 1) Optimization and linear algebra, and 2) Computer science applicable to large-scale scientific computing (including visualization, and the impact of the World Wide Web. Among the ideas discussed by your ever vigilant officers was a minisymposium of expository talks on orthogonal polynomials with heavy emphasis on the use of symbolic computation and links to online data bases as an aid to research. An associated session of research talks on the same topics was also proposed. A related idea is a session on handbooks (in a generalized sense, to include the World Wide Web). A session of expository or research talks related to Szego's work was suggested, appropriate since he was on the Stanford faculty. Another possibility is a session on Orthogonal Polynomials in Signal Processing. In general, Muldoon and Miller thought that a session of expository rather than research talks would be most useful to our membership, though more difficult to organize. They would greatly appreciate suggestions and advice from the SIAG-OS membership. A decision on the minisymposium topic or topics must be reached by late October 1996, for submission to the meeting organizers.

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