Box spline

The box spline associated with the -matrix of its directions (assumed non-zero) is the distribution defined inductively by

with the point evaluation at . This implies that , with the integral taken over the half-open unit -cube .

is (representable as) a piecewise-polynomial function on the linear hull of its directions, with support in the convex hull of its directions, its polynomial degree being equal to , its discontinuities on hyperplanes in spanned by its directions, and its smoothness across such a hyperplane determined by the number of directions lying in that hyperplane.

For and , is the uniform or cardinal B-spline. Correspondingly, the extant box spline theory (see [a1]) is a partial lifting of Schoenberg's cardinal spline theory [a2]. Its highlights include a study of the linear independence of the integer translates of a box spline (with integer directions), the shift-invariant spaces spanned by the integer translates of one or more box splines, the dimension of the space of polynomials contained in such a box spline space, the refinability of such box splines and the related subdivision schemes and discrete box splines.

References

[a1]	C. de Boor, K. Höllig, S. Riemenschneider, "Box splines" , Appl. Math. Sci. , 98 , Springer (1993)
[a2]	I.J. Schoenberg, "Cardinal spline interpolation" , CMBS , SIAM (1973)