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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク box spline ： ウィキペディア英語版 box spline 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 ($\backslash mathbb^d\; \backslash to\; \backslash mathbb$) defined for a set of vectors, $\backslash xi\; \backslash in\; \backslash mathbb^d$, usually gathered in a matrix $\backslash mathbf\; :=\; \backslash left(\backslash dots\; \backslash xi\_N\backslash right)$. When the number of vectors is the same as the dimension of the domain (i.e., $N\; =\; d$) then the box spline is simply the (normalized) indicator function of the parallelepiped formed by the vectors in $\backslash mathbf$: : Adding a new direction, $\backslash xi$, to $\backslash mathbf$, or generally when $N\; >\; d$, the box spline is defined recursively:〔 :$M\_(\backslash mathbf)\; =\; \backslash int\_0^1-\; t\; \backslash xi)\; \backslash ,\; t\}$. The box spline $M\_^N$ when projected down into $\backslash mathbb^d$. In this view, the vectors $\backslash xi\; \backslash in\; \backslash mathbf$ are the geometric projection of the standard basis in $\backslash mathbb^N$ (i.e., the edges of the hypercube) to $\backslash mathbb^d$. Considering tempered distributions a box spline associated with a single direction vector is a Dirac-like generalized function supported on $t\backslash xi$ for . Then the general box spline is defined as the convolution of distributions associated the single-vector box splines: :$M\_\; \backslash ast\; M\_\; \backslash dots\; \backslash ast\; M\_.$ 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 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.