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In the mathematical fields of numerical analysis and approximation theory, box splines are piecewise polynomial functions of several variables. Box splines are considered as a multivariate generalization of basis splines (B-splines) and are generally used for multivariate approximation/interpolation. Geometrically, a box spline is the shadow (X-ray) of a hypercube projected down to a lower-dimensional space. Box splines and simplex splines are well studied special cases of polyhedral splines which are defined as shadows of general polytopes. ==Definition== A box spline is a multivariate function () defined for a set of vectors, , usually gathered in a matrix . When the number of vectors is the same as the dimension of the domain (i.e., ) then the box spline is simply the (normalized) indicator function of the parallelepiped formed by the vectors in : : Adding a new direction, , to , or generally when , the box spline is defined recursively:〔 :. The box spline when projected down into . In this view, the vectors are the geometric projection of the standard basis in (i.e., the edges of the hypercube) to . Considering tempered distributions a box spline associated with a single direction vector is a Dirac-like generalized function supported on for . Then the general box spline is defined as the convolution of distributions associated the single-vector box splines: : 抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)』 ■ウィキペディアで「Box spline」の詳細全文を読む スポンサード リンク
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