Approximation error of the Lagrange reconstructing polynomial

The reconstruction approach [C.W. Shu, High-order weno schemes for convection-dominated problems, SIAM Rev. 51 (1) (2009) 82–126] for the numerical approximation of f′(x)f′(x) is based on the construction of a dual function h(x)h(x) whose sliding averages over the interval [x−12Δx,x+12Δx] are equal to f(x)f(x) (assuming a homogeneous grid of cell-size ΔxΔx). We study the deconvolution problem [A. Harten, B. Engquist, S. Osher, S.R. Chakravarthy, Uniformly high-order accurate essentially nonoscillatory schemes III, J. Comput. Phys. 71 (1987) 231–303] which relates the Taylor-polynomials of h(x)h(x) and f(x)f(x), and obtain its explicit solution, by introducing rational numbers τnτn defined by a recurrence relation, or determined by their generating function, gτ(x)gτ(x), related with the reconstruction pair of ex. 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.