Mathematics – Numerical Analysis
Scientific paper
2011-02-09
Mathematics
Numerical Analysis
Corrected typos
Scientific paper
In several applications, such as \tsc{weno} interpolation and reconstruction [Shu C.W.: {\em SIAM Rev.} {\bf 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 [M\"uhlbach G.: {\em Num. Math.} {\bf 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 [Carlini E., Ferretti R., Russo G.: {\em SIAM J. Sci. Comp.} {\bf 27} (2005) 1071-1091] for a general subdivision of the stencil $\{x_{i-M_-},...,x_{i+M_+}\}$ of $M+1:=M_-+M_++1$ distinct ordered points into $K_\mathrm{s}+1\leq M:=M_-+M_+>1$ (Neville) substencils $\{x_{i-M_-+k_\mathrm{s}},...,x_{i+M_+-K_\mathrm{s}+k_\mathrm{s}}\}$ ($k_\mathrm{s}\in\{0,...,K_\mathrm{s}\}$) all containing the same number of $M-K_\mathrm{s}+1$ 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 Y.Y., Shu C.W., Zhang M.P.: {\em Acta Math. Appl. Sinica} {\bf 25} (2009) 503-538].
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