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.: Num. Math. {\bf 31} (1978) 97--110] can be
generated by a general recurrence relation based on the existence of an
$1$-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.: 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 $ substencils
$\{x_{i-M_-+k_\mathrm{s}},...,x_{i+M_+-K_\mathrm{s}+k_\mathrm{s}}\}$
($k_\mathrm{s}\in\{0,...,K_\mathrm{s}\}$) containing the same number of points
$M-K_\mathrm{s}+1$ points but shifted by 1 cell, 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.: Acta Math. Appl. Sinica {\bf 25}
(2009) 503--538].