  We show that the presence of a side-attached state strongly modifies the
transmission through a one-dimensional double-barrier system in the window of
wavevectors around the Fano antiresonance. Specifically, the interplay between
the Fano interference and the size quantization inside the structure gives rise
to narrow resonant peaks in the transmission coefficient. The height of the
peaks may become close to unity (perfect transmission) even for an asymmetric
setup with strong barriers, where the transmission coefficient in the absence
of the Fano state is strongly suppressed at all other wavevectors. Thus, the
two types of the interference phenomena, each by itself leading to the
suppression of the transmission, conspire in a peculiar way to produce the
transparency enhancement.
