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Integrable Systems and Random Matrix Theory Seminar

Fast approximation on the real line
Monday, January 24, 2022
4:00-5:00 PM
ZOOM ID: 926 6491 9790 Off Campus Location
The original motivation for this work has been spectral methods for dispersive PDEs, whereby the underlying spatial domain is the whole real line. Seeking `good’ orthonormal systems on the real line, Marcus Webb and I have developed an overarching theory relating such systems which, in addition, have a tridiagonal differentiation matrix, to Fourier transforms of appropriately weighed orthogonal polynomials. Thus, for every Borel measure we obtain an orthonormal system in a Paley—Wiener space, which is dense in $L_2(\mathbb{R})$ iff the measure is supported on all of $\mathbb{R}$. In this talk I will introduce this theory, illustrate it by examples (in particular, Hermite and Malmquist—Takenaka functions) and, time allowing, take it in some of the following directions: - Characterisation of all such orthonormal systems whose first $n$ coefficients can be computed in $O(n \log n)$ operations;
- Sobolev-orthogonal systems of this kind;
- The emerging approximation theory on the real line.
Speaker(s): Arieh Iserles (University of Cambridge)
Building: Off Campus Location
Location: Virtual
Event Type: Workshop / Seminar
Tags: Mathematics
Source: Happening @ Michigan from Department of Mathematics