Temporally and Spatially Extended Star Formation in the Brightest
  Cluster Galaxy of MACS\,J0329.7$-$0211 at $z=0.45$: Implications for Stellar
  Growth
  Brightest cluster galaxies (BCGs), particularly those at the centers of
cool-core clusters, can exhibit star formation over spatial extents of up to
$\gtrsim$100\,kpc at inferred rates of up to $\gtrsim100\rm\,M_\sun\,yr^{-1}$.
Is their star formation also extended over time, as might be expected if
fuelled by cooling of the surrounding hot intracluster gas -- a residual
cooling flow -- as demonstrated hitherto only for the BCG in the Perseus
cluster? Here, to infer the formation history of relatively young stars in the
BCG of MACS\,J0329.7$-$0211, we fit model single-stellar-populations to the
spectral energy distributions (spanning near-UV to near-IR) measured along
different sightlines towards its young stellar population. Employing a Markov
Chain Monte Carlo method, we show that star formation in this BCG has persisted
at a relatively constant rate of $\sim2{\rm\,M_\sun\,yr^{-1}}$ (factors of
10--40 below the rates previously inferred using simpler methods and/or ad hoc
assumptions) over the past $\sim$400\,Myr, beyond which any star formation
falls below the observational detection threshold. Such persistent star
formation from a residual cooling flow can contribute up to $\sim$10\% of the
original stellar mass of this BCG if its progenitor was among the most massive
red nuggets known at $z\sim$2 having masses of $\sim1\times10^{11}\rm\,M_\sun$,
but only a few percent of its overall growth in stellar mass to
$\sim8\times10^{11}\rm\,M_\sun$ at $z=0.45$. Although constituting only a minor
pathway for the stellar growth of this BCG, persistent star formation from a
residual cooling flow can nevertheless contribute significantly to the enormous
number of globular clusters found around BCGs in the local Universe.
