  Higher-order accuracy (order of $k+1$ in the $L^2$ norm) is one of the well
known beneficial properties of the discontinuous Galerkin (DG) method.
Furthermore, many studies have demonstrated the superconvergence property
(order of $2k+1$ in the negative norm) of the semi-discrete DG method. One can
take advantage of this superconvergence property by post-processing techniques
to enhance the accuracy of the DG solution. A popular class of post-processing
techniques to raise the convergence rate from order $k+1$ to order $2k+1$ in
the $L^2$ norm is the Smoothness-Increasing Accuracy-Conserving (SIAC)
filtering. In addition to enhancing the accuracy, the SIAC filtering also
increases the inter-element smoothness of the DG solution. The SIAC filtering
was introduced for the DG method of the linear hyperbolic equation by Cockburn
et al. in 2003. Since then, there are many generalizations of the SIAC
filtering have been proposed. However, the development of SIAC filtering has
never gone beyond the framework of using spline functions (mostly B-splines) to
construct the filter function. In this paper, we first investigate the general
basis function (beyond the spline functions) that can be used to construct the
SIAC filter. The studies of the general basis function relax the SIAC filter
structure and provide more specific properties, such as extra smoothness, etc.
Secondly, we study the basis functions' distribution and propose a new SIAC
filter called compact SIAC filter that significantly reduces the original SIAC
filter's support size while preserving (or even improving) its ability to
enhance the accuracy of the DG solution. We show that the proofs of the new
SIAC filters' ability to extract the superconvergence and provide numerical
results to confirm the theoretical results and demonstrate the new finding's
good numerical performance.
