A new insight into the consistency of smoothed particle hydrodynamics
  In this paper the problem of consistency of smoothed particle hydrodynamics
(SPH) is solved. A novel error analysis is developed in $n$-dimensional space
using the Poisson summation formula, which enables the treatment of the kernel
and particle approximation errors in combined fashion. New consistency integral
relations are derived for the particle approximation which correspond to the
cosine Fourier transform of the classically known consistency conditions for
the kernel approximation. The functional dependence of the error bounds on the
SPH interpolation parameters, namely the smoothing length $h$ and the number of
particles within the kernel support ${\cal{N}}$ is demonstrated explicitly from
which consistency conditions are seen to follow naturally. As
${\cal{N}}\to\infty$, the particle approximation converges to the kernel
approximation independently of $h$ provided that the particle mass scales with
$h$ as $m\propto h^{\beta}$, with $\beta >n$. This implies that as $h\to 0$,
the joint limit $m\to 0$, ${\cal{N}}\to\infty$, and $N\to\infty$ is necessary
for complete convergence to the continuum, where $N$ is the total number of
particles. The analysis also reveals the presence of a dominant error term of
the form $(\ln {\cal{N}})^{n}/{\cal{N}}$, which tends asymptotically to
$1/{\cal{N}}$ when ${\cal{N}}\gg 1$, as it has long been conjectured based on
the similarity between the SPH and the quasi-Monte Carlo estimates.
