Mathematics Colloquia and Seminars

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It is well-known that classical polynomial interpolation with an N-th degree polynomial at N+1 equally spaced sample points can diverge even when the underlying function is analytic in a region containing the samples. This is known as the Runge phenomenon. In this talk we will present a family of schemes to construct interpolating polynomials that will converge no matter where the samples lie (the underlying function must satisfy a very mild smoothness condition; first derivative is not required for example). The convergence is, of course, only at limit points of the samples. The technique we present generalizes to any set of basis functions (not just polynomials) for which suitable Sobolev norms can be defined (for example, Laplace-Beltrami eigenfunctions on manifolds). Since our method places no constraints on the sample points, we can easily handle difficult approximation problems in high-dimensional spaces where the samples are restricted to complicated regions. Furthermore, our method exhibits "rapid local convergence" properties. By this we mean, that our interpolating polynomials will converge rapidly where the function is smooth, and more slowly where the function is singular. Note that such properties are more traditionally associated with wavelets. If time permits, we will show how these schemes can be used to construct meshless discretizations of differential and integral equations. This is joint work with M. Gu, H. Mhaskar, K. Raghuram, N. Somasundaram.

Note special time and room. There are two of the seminar series on this day.