Abstract
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.
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Communicated by Stephan Dahlke.
Supported by the National Natural Science Foundation of China (10871196) and by the Sofja Kovalevskaja Research Prize of Alexander von Humboldt Foundation awarded to Olga Holtz.
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Xu, Z. Multivariate F-splines and Fractional Box Splines. J Fourier Anal Appl 15, 723–738 (2009). https://doi.org/10.1007/s00041-009-9083-z
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DOI: https://doi.org/10.1007/s00041-009-9083-z