Sampling in reproducing kernel Banach spaces on Lie groups

We present sampling theorems for reproducing kernel Banach spaces on Lie groups. Recent approaches to this problem rely on integrability of the kernel and its local oscillations. In this paper, we replace these integrability conditions by requirements on the derivatives of the reproducing kernel and, in particular, oscillation estimates are found using derivatives of the reproducing kernel. This provides a convenient path to sampling results on reproducing kernel Banach spaces. Finally, these results are used to obtain frames and atomic decompositions for Banach spaces of distributions stemming from a cyclic representation. It is shown that this process is particularly easy, when the cyclic vector is a Gårding vector for a square integrable representation.