Approximation error of the Lagrange reconstructing polynomial

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.

💡 Research Summary

The paper provides a rigorous analytical treatment of the approximation error associated with Lagrange‑based polynomial reconstruction, a key component of modern high‑order finite‑volume and finite‑difference schemes for approximating derivatives on uniform grids. The authors begin by recalling the reconstruction framework introduced by Shu (2009): given cell‑averaged data (f_i) on a uniform mesh of spacing (\Delta x), one seeks a “reconstruction pair” ((f,h)) such that the sliding average of a dual function (h(x)) over the interval (