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Rapid Expansion in Orthogonal Polynomials

Monday, September 19, 2011
3:30PM – 5PM
POB 6.304

Dr. Arieh Iserles, Cambridge University

The computation of the first n terms of an expansion into orthogonal polynomials in O(n log n) operations is a long-standing challenge in computational mathematics. In the first part of the talk we describe such algorithms for the computation of expansions in ultrasperical (a.k.a. Gegenbauer) polynomials. This proceeds in three somewhat counterintuitive steps. Firstly, expansion coefficients are expressed as an infinite linear combination of derivatives. Secondly, using the Cauchy theorem, this is converted into an integral transform with a hypergeometric kernel.

Finally, using a serendipitous transformation of the kernel into a rapidly-convergent function, the integral transform is computed as a finite linear combination of a discrete Fourier transform of the underlying function along a Bernstein ellipse.

In the second part of the talk we generalize the first two stages to arbitrary orthogonal polynomial systems supported by compact real intervals, as well as to polynomials and Laurent polynomials orthogonal on the unit circle.

This is joint work with María José Cantero.

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