This talk gives an outline of (mostly unfinished) work done collaboratively while on sabbatical in semester 2 last year. Join me as we travel through the USA, Germany, Belgium and Austria. Your guide will share off-the-beaten-track highlights such as quaternionic splines, prolate shift systems, higher-dimensional Hardy, Paley-Wiener and Bernstein spaces, the Clifford Fourier transform, multidimensional prolates, and a Jon Borwein-inspired optimization-based approach to the construction of multidimensional wavelets. Breakfast not included.

This forum is a follow-on from the seminar that Professor Willis gave three weeks prior, on maths that seems too good to be true; and his ideas for incorporating the surprisingly and enlivening into what and how we teach: he gave as exemplars the miracles of Pythagoreans triples and eigenvalues. A question raised in the discussion at that seminar was if/how might we use assessment to encourage the kinds of learning we would like. This forum will be an opportunity to further that conversation.

Jeff, Andrew and Massoud have each kindly agreed to give us 5 minute presentations relating to the latter year maths courses that they have recently been teaching, to get our forum started. Jeff may speak on his developments in his new course on Fourier methods, Andrew will talk about some of the innovations that were introduced into Topology in the last few offerings which he has been using and further developing, and Massoud has a range of OR courses he might speak about.

Everyone is encouraged to share examples of their own practice or ideas that they have that may be of interest to others.

The classical prolate spheroidal wavefunctions (prolates) arise when solving the Helmholtz equation by separation of variables in prolate spheroidal coordinates. They interpolate between Legendre polynomials and Hermite functions. In a beautiful series of papers published in the Bell Labs Technical Journal in the 1960's, they were rediscovered by Landau, Slepian and Pollak in connection with the spectral concentration problem. After years spent out of the limelight while wavelets drew the focus of mathematicians, physicists and electrical engineers, the popularity of the prolates has recently surged through their appearance in certain communication technologies. In this talk we outline some developments in the sampling theory of bandlimited signals that employ the prolates, and the construction of bandpass prolate functions.
This is joint work with Joe Lakey (New Mexico State University)

The classical prolate spheroidal wavefunctions (prolates) arise when solving the Helmholtz equation by separation of variables in prolate spheroidal coordinates. They interpolate between Legendre polynomials and Hermite functions. In a beautiful series of papers published in the Bell Labs Technical Journal in the 1960's, they were rediscovered by Landau, Slepian and Pollak in connection with the spectral concentration problem. After years spent out of the limelight while wavelets drew the focus of mathematicians, physicists and electrical engineers, the popularity of the prolates has recently surged through their appearance in certain communication technologies. In this talk we discuss the remarkable properties of these functions, the ``lucky accident'' which enables their efficient computation, and give details of their role in the localised sampling of bandlimited signals.