  The reconstruction approach [Shu C.W.: {\em SIAM Rev.} {\bf 51} (2009)
82--126] for the numerical approximation of $f'(x)$ is based on the
construction of a dual function $h(x)$ whose sliding averages over the interval
$[x-\tfrac{1}{2}\Delta x,x+\tfrac{1}{2}\Delta x]$ are equal to $f(x)$ (assuming
an homogeneous grid of cell-size $\Delta x$). We study the deconvolution
problem [Harten A., Engquist B., Osher S., Chakravarthy S.R.: {\em J. Comp.
Phys.} {\bf 71} (1987) 231--303] which relates the Taylor polynomials of $h(x)$
and $f(x)$, and obtain its explicit solution, by introducing rational numbers
$\tau_n$ defined by a recurrence relation, or determined by their generating
function, $g_\tau(x)$, related with the reconstruction pair of ${\rm e}^x$. We
then apply these results to the specific case of Lagrange-interpolation-based
polynomial reconstruction, and determine explicitly the approximation error of
the Lagrange reconstructing polynomial (whose sliding averages are equal to the
Lagrange interpolating polynomial) on an arbitrary stencil defined on a
homogeneous grid.
