Calculating areas of box spline surfaces (Q1902406)

After a short introduction to box splines, the authors give upper and lower bounds for the area of a surface given by a box spline approximation \(S = \Sigma a (i,j) B_v (x - i, y-j)\) where \(v\) is a multi-index. They define recursively finer approximations and are interested in the error committed when the multi-index box spline \(B_v\) is replaced by the basic box spline \(B_u(x,y) = \int_{- 1/2}^{1/2}B(x + t, y + t) dt\) where \(B\) is the characteristic function of the unit square centered at the origin. They show monotone quadratic approximation of the area both from above and from below, making for a reliable algorithm.