Box spline

The box spline $ M _ \Xi $ associated with the $ ( s \times n ) $- matrix $ \Xi = [ \xi _ {1} \dots \xi _ {n} ] $ of its directions $ \xi _ {i} $( assumed non-zero) is the distribution defined inductively by

$$ M _ {[ \Xi, \zeta ] } = \int\limits _ { 0 } ^ { 1 } {M _ \Xi ( \cdot - t \zeta ) } {d t } $$

with $ M _ {[ ] } $ the point evaluation at $ 0 \in \mathbf R ^ {s} $. This implies that $ M _ \Xi \phi = \int {\phi ( \Xi t ) } {d t } $, with the integral taken over the half-open unit $ n $- cube $ [ 0,1 ) ^ {n} $.

$ M _ \Xi $ is (representable as) a piecewise-polynomial function on the linear hull $ { \mathop{\rm ran} } \Xi $ of its directions, with support in the convex hull of its directions, its polynomial degree being equal to $ s - { \mathop{\rm dim} } { \mathop{\rm ran} } \Xi $, its discontinuities on hyperplanes in $ { \mathop{\rm ran} } \Xi $ spanned by its directions, and its smoothness across such a hyperplane determined by the number of directions lying in that hyperplane.

For $ s = 1 $ and $ \Xi = [ 1 \dots 1 ] $, $ M _ \Xi $ 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)