Box-splines orthogonal projections

Let P be orthogonal projection on B-splines of degree r-1 with equally spaced knots. Sweldens and Piessens proved that P(x^r)-x^r is Bernoulli polynomial. We generalize Sweldens ans Piessens's result for box-splines. It gives the opportunity to define the seminorm of Sobolev space in terms of the asymptotic formula for the error in orthogonal projection.