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Orthogonal systems with a skew-symmetric differentiation matrix

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For certain time-dependent PDEs, the norm of the solution as time progresses necessarily decays, or is preserved, e.g. the diffusion, Schrödinger, or nonlinear advection equations, a property due to the skew-hermitian nature of the differentiation operator. For a numerical solution of these PDEs, these properties of the underlying analytical solutions can be perfectly respected if the matrix representing differentiation in your discretisation is skew-hermitian too. In this talk, we characterise all systems of orthogonal functions on L2® such that the differentiation matrix for an expansion in these functions is real, skew-symmetric, tridiagonal and irreducible, accomplished by interesting links between orthogonal polynomials, the Fourier transform, and Paley-Wiener band-limited function spaces. This is joint work with Arieh Iserles (Cambridge), with a preprint available at http://www.damtp.cam.ac.uk/user/na/NA_papers/NA2018_02.pdf.

This talk is part of the Cambridge Analysts' Knowledge Exchange series.

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