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Masterclass: singular integrals and orthogonal polynomials

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CATW03 - Computational complex analysis

Orthogonal polynomials are fundamental tools in numerical methods, including for numerical methods 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 that converge spectrally fast (faster than any algebraic power), uniformly in the complex plane. Closed formulae for Cauchy transforms on more complicated geometries are derivable using the Plemelj lemma. These techniques extend to other singular integrals such as those with logarithmic kernels.

We will demonstrate these results in Julia using ApproxFun.jl and SingularIntegralEquations.jl.

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

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