Representation of the Lagrange reconstructing polynomial by combination of substencils

The Lagrange reconstructing polynomial [C.W. Shu, High-order WENO schemes for convection-dominated problems, SIAM Rev. 51 (1) (2009) 82–126] of a function f(x)f(x) on a given set of equidistant (Δx=const) points {xi+ℓΔx;ℓ∈{−M−,…,+M+}} is defined as the polynomial whose sliding (with xx) averages on [x−12Δx,x+12Δx] are equal to the Lagrange interpolating polynomial of f(x)f(x) on the same stencil [G.A. Gerolymos, Approximation error of the Lagrange reconstructing polynomial, J. Approx. Theory 163 (2) (2011) 267–305. http://dx.doi.org/10.1016/j.jat.2010.09.007]. We first study the fundamental functions of Lagrange reconstruction, then show that these polynomials have only real and distinct roots, which are never located at the cell-interfaces (half-points) xi+n12Δx (n∈Zn∈Z), and obtain several identities. Using these identities, we show that there exists a unique representation of the Lagrange reconstructing polynomial on {i−M−,…,i+M+}{i−M−,…,i+M+} as a combination of the Lagrange reconstructing polynomials on Neville substencils [E. Carlini, R. Ferretti, G. Russo, A WENO large time-step scheme for Hamilton–Jacobi equations, SIAM J. Sci. Comput. 27 (3) (2005) 1071–1091], with weights which are rational functions of ξξ (x=xi+ξΔxx=xi+ξΔx) [Y.Y. Liu, C.W. Shu, M.P. Zhang, On the positivity of the linear weights in WENO approximations, Acta Math. Appl. Sin. 25 (3) (2009) 503–538], and give an analytical recursive expression of the weight-functions. We show that all of the poles of the rational weight-functions are real, and that there can be no poles at half-points. We then use the analytical expression of the weight-functions, combined with the factorization of the fundamental functions of Lagrange reconstruction, to obtain a formal proof of convexity (positivity of the weight-functions) in the neighborhood of ξ=12, iff all of the substencils contain either point ii or point i+1i+1 (or both).