A general recurrence relation for the weight-functions in Mühlbach–Neville–Aitken representations with application to WENO interpolation and differentiation

In several applications, such as WENO interpolation and reconstruction [C.W. Shu, SIAM Rev. 51 (2009) 82–126], we are interested in the analytical expression of the weight-functions which allow the representation of the approximating function on a given stencil (Chebyshev-system) as the weighted combination of the corresponding approximating functions on substencils (Chebyshev-subsystems). We show that the weight-functions in such representations [G. Mühlbach, Numer. Math. 31 (1978) 97–110] can be generated by a general recurrence relation based on the existence of a 1-level subdivision rule. As an example of application we apply this recurrence to the computation of the weight-functions for Lagrange interpolation [E. Carlini, R. Ferretti, G. Russo, SIAM J. Sci. Comput. 27 (2005) 1071–1091] for a general subdivision of the stencil {xi-M-,…,xi+M+}{xi-M-,…,xi+M+} of M+1≔M-+M++1M+1≔M-+M++1 distinct ordered points into Ks+1⩽M≔M-+M+>1Ks+1⩽M≔M-+M+>1 (Neville) substencils {xi-M-+ks,…,xi+M+-Ks+ks}{xi-M-+ks,…,xi+M+-Ks+ks} (ks∈{0,…,Ks}ks∈{0,…,Ks}) all containing the same number of M-Ks+1M-Ks+1 points but each shifted by 1 cell with respect to its neighbour, and give a general proof for the conditions of positivity of the weight-functions (implying convexity of the combination), extending previous results obtained for particular stencils and subdvisions Liu (2009) [Y.Y. Liu, C.W. Shu, M.P. Zhang, Acta Math. Appl. Sin. 25 (2009) 503–538]. Finally, we apply the recurrence relation to the representation by combination of substencils of derivatives of arbitrary order of the Lagrange interpolating polynomial.