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    Conference on Experimental and Constructive Mathematics (CECM) 99 -- Analysis Day

    (Wednesday) August 4, 1999--Room K9509, Burnaby Campus, Simon Fraser University

Schedule of Talks
9:00 - 9:45 Heinz Bauschke, Okanagan University College
Accelerating the Convergence of the Method of Alternating Projections

The powerful von Neumann-Halperin method of alternating projections (MAP) is an algorithm for determining the best approximation to any given point in a Hilbert space from the intersection of a finite number of subspaces.

In this talk, I consider an accelerated version of this algorithm originally suggested (but not analyzed) by Gubin-Polyak-Raik. I explain when and why this method really accelerates.

(Based on joint work with Frank Deutsch, Hein Hundal, and Sung-Ho Park.)

9:45 - 10:30 Adrian Lewis, University of Waterloo
Ill-conditioned Systems of Inequalities

Convex processes provide an elegant tool for studying both cones and inequality systems. I will show that the smallest linear perturbation making a convex process nonsurjective has norm the reciprocal of the norm of the inverse process. This generalizes the fundamental property of the condition number of a linear map. I will then apply this result to strengthen a theorem of Renegar measuring the size of perturbation necessary to make an inequality system inconsistent. This measure is fundamental in the complexity of corresponding interior point methods.

10:30 - 10:45 Coffee Break
10:45 - 11:30 Ivaylo Kortezov, Simon Fraser University
Game Approach and Fragmentability Properties of Function Spaces

Using some theorems of P. Kenderov and W. Moors for characterizing fragmentability and sigma-fragmentability of topological (in particular Banach) spaces, we prove fragmentability of certain classes of function spaces and show some preservation results under topological operations.

11:30 - 12:15 Joel Kamnitzer, University of Waterloo
Central Binomial Sums

The following interesting summations are well known: \sum _{n=0}^\infty \frac{1}{2n choose n}{n^2} = \Zeta(2)/3 and
\sum _{n=0}^\infty \frac{(-1)^{n+1}}{2n choose n}{n^3} = \frac{2}{5} \Zeta(3)
with the latter playing a key roal in Apery's proof that \Zeta(3) is irrational.

Our goal is to extend these results by expressing the sum S(k) = \sum _{n=0}^\infty \frac{1}{2n choose n}{n^k} in terms of Zeta values. Along the way we are lead to Multiple Claussen Values, which prove to be an interesting area of study in their own right.

12:15 - 1:30 Lunch Break
1:30 - 3:30 Xianfu Wang, Simon Fraser University
Fine and Pathological Properties of Subdifferentials (Thesis Defence)
Please note: Thesis defence takes place in Room AQ4100.
3:30 - 4:15 Roland Girgensohn, National Research Center for Environment and Health, Institute of Biomathematics and Biometry, Neuherberg, Germany.
Schauder Bases Consisting of Orthogonal Polynomials

We construct Schauder bases for the space $C[-1,1]$ which consist of algebraic polynomials orthogonal with respect to a given weight function (e.g., Jacobi weights). The polynomials are of minimal degree, that is, for given $\varepsilon>0$ there is such a basis~$\{p_\mu\}_{\mu=0}^\infty$ with $\deg p_\mu \le (1+\varepsilon)\,\mu$ for all $\mu \in I\!\!N_0$.

When constructing orthogonal Schauder bases, one has to estimate certain Lebesgue constants. Here we use ideas from wavelet theory for this purpose. The space $C[-1,1]$ is decomposed into a sequence of finite-dimensional subspaces, consisting of polynomials and orthogonal
to each other, which can be thought of as spaces of wavelets: each space is the span of (generalized) translates of a certain generating polynomial. In this way, the estimation of Lebesgue constants can be reduced to estimates for these finite-dimensional subspaces.

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