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Orthogonal polynomials, singular integrals, and solving Riemann–Hilbert problems: Lecture 2

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WHT - Bringing pure and applied analysis together via the Wiener-Hopf technique, its generalisations and applications

Orthogonal polynomials are fundamental tools in numerical methods, including for singular integral equations. A known result is that Cauchy transforms of weighted orthogonal polynomials satisfy the same three-term recurrences as the orthogonal polynomials themselves for n > 0. This basic fact leads to extremely effective schemes of calculating singular integrals and discretisations of singular integral equations that converge spectrally fast (faster than any algebraic power). Applications considered include matrix Riemann–Hilbert problems on contours consisting of interconnected line segments and Wiener–Hopf problems. This technique is extendible to calculating singular integrals with logarithmic kernels, with applications to Green’s function reduction of PDEs such as the Helmholtz equation.  

Using novel change-of-variable formulae, we will adapt these results to tackle singular integral equations on more general smooth arcs, geometries with corners, and Wiener–Hopf problems whose solutions only decay algebraically.

This talk is part of the Isaac Newton Institute Seminar Series series.

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