Computer Aided Geometric Design 3 (1986) 185-191 185 North-Holland On the piecewise structure of discrete box splines * Wolfgang DAHMEN Fakultiit./'fir Mathematik, Universitiit Bielefeld, 4800 Bielefela~ West Germany Charles A. MICCHELLI IBM T.J. Watson Research Center, P.O. Box 218, Yorktown Heights, NY 10598, U.S.A. Received 5 November 1985 Revised 8 April 1986 Abstract. In this note we give an explicit example where the discrete box spline is shown to have a non-polynomial piecewise structure. We briefly point out how the example relates to some general results which are useful in describing the correct piecewise nature of such splines. Keywords. Discrete box splines, discrete truncated powers, subdivision algorithms, difference operators In [Cohen et al. '84, Dahmen and Micchelli '841 discrete box splines were introduced as an important tool in discussing subdivision algorithms for continuous box spline surfaces. While in the univariate case the discrete B-splines are piecewise polynomials, it has been mistakenly asserted in [Cohen et al. '84] that the same is true for multivariate box splines. The purpose of this paper is to show that in general this is not the case. In particular, we give an explicit example where the discrete box spline has non-polynomial piecewise structure. In addition we relate the example to some general results which appear to be useful in delineating the correct piecewise nature of such splines. A complete characterization of the piecewise structure of discrete box splines with proofs for the general case will appear elsewhere. To explain our example we recall some definitions from [Dahmen and Micchelli '84]. For X= {xt,..., x") c Z s \ (0} the discrete truncated power t(. [ X) is defined by requiring that the relation ~_, f(a) t(alS)= Y'~ f(fll xl + -'- +flnX n) holds for any f: Z s ---, R vanishing except on a finite number of lattice points. In the special bivariate case, X = {(1, 0), (0, 1), (1, 1), (- 1, 1)}, a direct calculation given in the Appendix yields for any a = (a 1, a2) ~ Z 2 'f,(a)=½(a2+ l)(%+ 2), al>~a2>_-O, f2(a)----- 1 [2(al + a 2 + 2)2 -- 4al(al + 1)- 1 + (- 1)a1+"2], t(al X) = 0 ~<a 1 ~< Ot2, (2) f3(a) = ~ [2(oq + a 2 + 2) 2- 1 + (-1)'1+'2], --a 2~<Ot 1~<0, a 2>/0, ,0, otherwise. * This work was partially supported by NATO Grant DJ RG 639/84. 0167-8396/86/$3.50 © 1986, Elsevier Science Publishers B.V. (North-Holland)