  Convolving the output of Discontinuous Galerkin (DG) computations with
symmetric Smoothness-Increasing Accuracy-Conserving (SIAC) filters can improve
both smoothness and accuracy. To extend convolution to the boundaries, several
one-sided spline filters have recently been developed. We interpret these
filters as instances of a general class of position-dependent spline filters
that we abbreviate as PSIAC filters. These filters may have a non-uniform knot
sequence and may leave out some B-splines of the sequence.
  For general position-dependent filters, we prove that rational knot sequences
result in rational filter coefficients. We derive symbolic expressions for
prototype knot sequences, typically integer sequences that may include repeated
entries and corresponding B-splines, some of which may be skipped. Filters for
shifted or scaled knot sequences are easily derived from these prototype
filters so that a single filter can be re-used in different locations and at
different scales. Moreover, the convolution itself reduces to executing a
single dot product making it more stable and efficient than the existing
approaches based on numerical integration. The construction is demonstrated for
several established and one new boundary filter.
