  Using the reconstruction technique with an auxiliary field, we investigate
which $F(R)$ gravities can produce the matter bounce cosmological solutions.
Owing to the specific functional form of the matter bounce Hubble parameter,
the reconstruction technique leads, after some simplifications, to the same
Hubble parameter as in the matter bounce scenario. Focusing the study to the
large and small cosmic time $t$ limits, we were able to find which $F(R)$
gravities can generate the matter bounce Hubble parameter. In the case of small
cosmic time limit, which corresponds to large curvature values, the $F(R)$
gravity is $F(R)\sim R+\alpha R^2$, which is an inflation generating gravity,
and at small curvature, or equivalently, large cosmic time, the $F(R)$ gravity
generating the corresponding limit of the matter bounce Hubble parameter, is
$F(R)\sim \frac{1}{R}$, a gravity known to produce late-time acceleration. Thus
we have the physically appealing picture in which a Jordan frame $F(R)$ gravity
that imitates the matter bounce solution at large and small curvatures, can
generate Starobinsky inflation and late-time acceleration. Moreover, the scale
factor corresponding to the reconstruction technique coincides almost
completely to the matter bounce scenario scale factor, when considered in the
aforementioned limiting curvature cases. This is scrutinized in detail, in
order to examine the validity of the reconstruction method in these limiting
cases, and according to our analysis, exact agreement is achieved.
