On dual molecules and convolution-dominated operators

Abstract

We show that sampling or interpolation formulas in reproducing kernel Hilbert spaces can be obtained by reproducing kernels whose dual systems form molecules, ensuring that the size profile of a function is fully reflected by the size profile of its sampled values. The main tool is a local holomorphic calculus for convolution-dominated operators, valid for groups with possibly non-polynomial growth. Applied to the matrix coefficients of a group representation, our methods improve on classical results on atomic decompositions and bridge a gap between abstract and concrete methods.

Interpolation in Wavelet Spaces and the HRT-Conjecture

“…The fact that the wavelet spaces have a reproducing kernel Hilbert space structure originally appeared in the influential paper [25]. Since then, it has been used in both special cases [1] and in the general setting [40].…”