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) |
Box spline. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Box_spline&oldid=46141