On dual molecules and convolution-dominated operators

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. (C) 2021 Elsevier Inc. All rights reserved.

Automated summary

In this study, it is demonstrated 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 methods utilized include a local holomorphic calculus for convolution-dominated operators, applicable to groups with possibly non-polynomial growth. When applied to the matrix coefficients of a group representation, these methods improve upon classical results on atomic decompositions and bridge the gap between abstract and concrete methods.