View source for Box spline

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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 [[#References|[a1]]]) is a partial lifting of Schoenberg's cardinal spline theory [[#References|[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====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  C. de Boor,   K. Höllig,   S. Riemenschneider,   "Box splines" , ''Appl. Math. Sci.'' , '''98''' , Springer  (1993)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  I.J. Schoenberg,   "Cardinal spline interpolation" , ''CMBS'' , SIAM  (1973)</TD></TR></table>