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)
How to Cite This Entry:
Box spline. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Box_spline&oldid=46141
This article was adapted from an original article by C. de Boor (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article