Polynomial interpolation on interlacing rectangular grids

In this talk we review some of the remarkable properties of Padua points and related point sets consisting of pairs of interlacing rectangular grids.

In particular, Padua points, defined in the domain $[-1,1]^2$, are unisolvent for polynomial interpolation of full degree $N$.
The Lebesgue constant grows with minimal order $O(\log^2(N))$ and the associated cubature rule has degree of precision $2N-1$ with respect to the Chebyshev weighting.
Similar properties have been established by Morrow and Patterson, Xu, and Erb et al. for similar pairs of interlacing rectangular grids, with respect to suitable spaces of polynomials that are no longer full, but in some cases close to full.
In all these grids, the points have some kind of Chebyshev spacing in each coordinate direction.

We will then go on to focus purely on unisolvence and study the unisolvence of interlacing pairs of rectangular grids in which the spacing of the points in each coordinate direction is arbitrary.
We will break this problem down using a combination of tensor-product interpolation, Newton interpolation, and a property of divided difference matrices.

Michael Floater received a PhD in mathematics from Oxford University in 1988.
He later worked for several years at SINTEF in Oslo, Norway, on various industrial projects, many of which involved geometric modelling.
He has been at the University of Oslo since 2003, currently in the Department of Mathematics. He has published 90 refereed papers and edited 4 books.
His main research in recent years has been focused on numerical methods for interpolation and approximation.