  This paper considers different facets of the interplay between reproducing
kernel Hilbert spaces (RKHS) and stable analysis/synthesis processes: First, we
analyze the structure of the reproducing kernel of a RKHS using frames and
reproducing pairs. Second, we present a new approach to prove the result that
finite redundancy of a continuous frame implies atomic structure of the
underlying measure space. Our proof uses the RKHS structure of the range of the
analysis operator. This in turn implies that all the attempts to extend the
notion of Riesz basis to general measure spaces are fruitless since every such
family can be identified with a discrete Riesz basis. Finally, we show how the
range of the analysis operators of a reproducing pair can be equipped with a
RKHS structure.
