  An interesting topic in compressive sensing concerns problems of sensing and
recovering signals with sparse representations in a dictionary. In this note,
we study conditions of sensing matrices A for the L1-synthesis method to
accurately recover sparse, or nearly sparse signals in a given dictionary D. In
particular, we propose a dictionary based null space property (D-NSP) which, to
the best of our knowledge, is the first sufficient and necessary condition for
the success of the L1 recovery. This new property is then utilized to detect
some of those dictionaries whose sparse families cannot be compressed
universally. Moreover, when the dictionary is full spark, we show that AD being
NSP, which is well-known to be only sufficient for stable recovery via
L1-synthesis method, is indeed necessary as well.
