Multivariate F-splines and Fractional Box Splines

  • Zhiqiang Xu


We introduce multivariate F-splines, including multivariate F-truncated powers T f (⋅|M) and F-box splines B f (⋅|M). The classical multivariate polynomial splines and multivariate E-splines can be considered as a special case of multivariate F-splines. We document the main properties of T f (⋅|M) and B f (⋅|M). Using T f (⋅|M), we extend fractional B-splines to fractional box splines and show that these functions satisfy most of the properties of the traditional box splines.

Our work unifies and generalizes results due to Dahmen-Micchelli, de Boor-Höllig, Ron and Unser-Blu, and also presents a new tool for computing the integration over polytopes.


Box splines Multivariate truncated powers Fractional B-splines 

Mathematics Subject Classification (2000)

41A15 65D07 42C40 


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Copyright information

© Birkhäuser Boston 2009

Authors and Affiliations

  1. 1.LSEC, Inst. Comp. Math., Academy of Mathematics and System SciencesChinese Academy of SciencesBeijingChina

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