Approximation to scattered data, using translates of one
basis function such as the Gaussian kernel, the thin plate spline, or the
multiquadric.
Multivariate polynomial interpolation; specifically, the least solution
for that problem.
Wavelet systems and Gabor systems: primarily fiberization techniques
for such sets (useful for the study of the Riesz basis property, and the frame
property of such systems).
Shift-invariant spaces. These are spaces of functions that
are invariant under integer translations. I am interested in several topics
in this area including the approximation orders and approximation schemes
from such spaces, and shift-invariant bases for such spaces. More specific
topics here include box splines , and refinable spaces (that
are used for wavelet constructions).
Dimension formulae for joint kernels of commuting linear operators. This is a
topic that started at box spline theory, and has connections to ideal theory
and matroids.