Blossoming begets $B$-spline bases built better by $B$-patches

The concept of symmetric recursive algorithm leads to new, s-dimensional spline spaces. We present a general scheme for constructing a collection of multivariate B-splines with $k - 1$ continuous derivatives whose linear span contains all polynomials of degree at most k. This scheme is different from the one developed earlier by Dahmen and Micchelli and, independently, by Höllig, which was based on combinatorial principles and the geometric interpretation of the B-spline. The new spline space introduced here seems to offer possibilities for economizing the computation for evaluating linear combinations of B-splines.