View source for Box spline

The box spline <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110800/b1108001.png" /> associated with the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110800/b1108002.png" />-matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110800/b1108003.png" /> of its directions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110800/b1108004.png" /> (assumed non-zero) is the distribution defined inductively by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110800/b1108005.png" /></td> </tr></table>

with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110800/b1108006.png" /> the point evaluation at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110800/b1108007.png" />. This implies that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110800/b1108008.png" />, with the integral taken over the half-open unit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110800/b1108009.png" />-cube <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110800/b11080010.png" />.

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110800/b11080011.png" /> is (representable as) a piecewise-polynomial function on the linear hull <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110800/b11080012.png" /> of its directions, with support in the convex hull of its directions, its polynomial degree being equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110800/b11080013.png" />, its discontinuities on hyperplanes in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110800/b11080014.png" /> spanned by its directions, and its smoothness across such a hyperplane determined by the number of directions lying in that hyperplane.

For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110800/b11080015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110800/b11080016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110800/b11080017.png" /> 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>