  We consider the discontinuous Galerkin method for hyperbolic conservation
laws, with some particular attention to the linear acoustic equation, using
Bernstein polynomials as local bases. Adapting existing techniques leads to
optimal-complexity computation of the element and boundary flux terms. The
element mass matrix, however, requires special care. In particular, we give an
explicit formula for its eigenvalues and exact characterization of the
eigenspaces in terms of the Bernstein representation of orthogonal polynomials.
We also show a fast algorithm for solving linear systems involving the element
mass matrix to preserve the overall complexity of the DG method. Finally, we
present numerical results investigating the accuracy of the mass inversion
algorithms and the scaling of total run-time for the function evaluation needed
in DG time-stepping.
