  In this paper, we solve a separation of singularities problem in the Bergman
space. More precisely, we show that if $P\subset \mathbb{C}$ is a convex
polygon which is the intersection of $n$ half planes, then the Bergman space on
$P$ decomposes into the sum of the Bergman spaces on these half planes. The
result applies to the characterization of the reachable space of the
one-dimensional heat equation on a finite interval with boundary controls. We
prove that this space is a Bergman space of the square which has the given
interval as a diagonal. This gives an affirmative answer to a conjecture raised
in [HKT20].
