Speaker: Arieh Iserles (University of Cambridge)
Abstract: Many invariants of time-evolving PDEs, e.g. mass and some Hamiltonians, can be formulated as a bilinear form. In this talk we are concerned with formal orthogonal systems on the real line (and, by tensorisation, in $\mathbb{R}^d$) with respect to bilinear forms and with a tridiagonal differentiation matrix. The theory is virtually complete for $L_2$ and Sobolev inner products: using a Fourier transform we show that such systems are in a one-to-one relationship with determinate Borel measures and that their closure is a Paley–Wiener space. We provide several examples, commencing from the familiar Hermite functions. We also characterise all such systems that can be computed fast – for $L_2$ orthogonality using Fast Fourier/Cosine/Sine Transform, for Sobolev orthogonality the above in tandem with a narrow-banded matrix of connection coefficients. We conclude with preliminary results on systems that are orthogonal with respect to a bilinear form generated by the Hamiltonian of a linear Schrödinger equation. This is joint work with Marcus Webb.
Meeting ID:: 889 4601 2321
Access Code: 179101
