Multivariate B-Splines — Recurrence Relations and Linear Combinations of Truncated Powers

Abstract

In [3] C.de Boor suggests the following definition of multivariate B-splines as an analogous notion to the well known univariate B-splines introduced by H. B. Curry and I. J. Schoenberg [l]. Let a be an n-simplex of unit volume then “for $\underline x {\kern 1pt} \varepsilon {\kern 1pt} I{R^s},s \leqslant n$, the (n-s)-dimensional volumes

$${M_\sigma }\left( {\underline x } \right): = vo{\operatorname{l} _{n - s}}\left( {\left\{ {\underline u \;\varepsilon \;\sigma :\underline u \left| {_\mathbb{R}s} \right. = \underline x } \right\}} \right)$$

((1.1))

define a nonnegative and locally supported function which may be expected to be a smooth piecewise polynomial of total degree n-s. Indeed, this is confirmed by alternative represen-tations in terms of multivariate truncated powers and by recurrence relations for ${M_\sigma }(\underline x )$ which were recently derived in [2], [4]. These results suggest that the multivariate B-splines may become a useful tool for practical applications and may also reveal new aspects of a multivariate constructive function theory.