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.

中文翻译：

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关于双分子和卷积支配的算子

我们证明，通过重现其双系统形成分子的内核，可以确保重现内核希尔伯特空间中的采样或内插公式，从而确保函数的大小轮廓可以完全由其采样值的大小轮廓反映出来。主要工具是用于卷积控制算子的局部全纯演算，适用于可能具有非多项式增长的群。将这些方法应用于群表示的矩阵系数后，我们改进了原子分解的经典结果，并弥合了抽象方法与具体方法之间的差距。