On the piecewise structure of discrete box splines (Q1087300)

Discrete box splines were introduced by \textit{E. Cohen}, \textit{T. Lyche} and \textit{R. Riesenfeld} [ibid. 1, 131-148 (1984; Zbl 0567.65004)] and the authors [ibid. 1, 115-129 (1984; Zbl 0581.65011)]. Perhaps the tacit assumption about the linearity of the backward difference operator \(v\to \nabla_ V\) of E. Cohen et. al. (loc. cit.) led to the conclusion that the discrete box splines have piecewise polynomial structure. The authors show that the foregoing conclusion is not true and they support their claim by an illustration.