Scalar multivariate subdivision schemes and box splines

We study convergent scalar d-variate subdivision schemes satisfying sum rules of order k ∈ N, with dilation matrix 2I . Using the results of Möller and Sauer in [18], stated for general expanding dilation matrices, we characterize the structure of the mask symbols of such schemes by showing that they must be linear combinations of shifted box spline generators of a quotient polynomial ideal J . The directions of the corresponding box splines are θ ∈ {0, 1} \ {(0, . . . , 0)}. The quotient ideal J , as shown in [18], is determined by the given order of the sum rules or, equivalently, by the order of the Strang–Fix conditions. Our results open a way to a systematic study of subdivision schemes. For example, in the bivariate case, if the mask symbol of any convergent subdivision scheme is in J , then the mask is an affine combination of smoothed versions of three-directional box splines. Many special cases, including affine combinations of convergent schemes, can be looked at this way; see, e.g., [7] and the references given therein. As in the univariate case, this characterization seems to be the proper way of matching the smoothness, as determined in [1], of the box spline building blocks with the order of polynomial reproduction of the corresponding scheme. Due to the interaction of the building blocks, the convergence and smoothness, however, are usually destroyed, if several convergent schemes are combined in this way. We illustrate our results with several examples.