  We derive an extension of the Walnut-Daubechies criterion for the
invertibility of frame operators. The criterion concerns general reproducing
systems and Besov-type spaces. As an application, we conclude that $L^2$ frame
expansions associated with smooth and fast-decaying reproducing systems on
sufficiently fine lattices extend to Besov-type spaces. This simplifies and
improves recent results on the existence of atomic decompositions, which only
provide a particular dual reproducing system with suitable properties. In
contrast, we conclude that the $L^2$ canonical frame expansions extend to many
other function spaces, and, therefore, operations such as analyzing using the
frame, thresholding the resulting coefficients, and then synthesizing using the
canonical dual frame are bounded on these spaces.
