It was observed in [
4] that the Hilbert transform of the univariate B-spline preserves the B-spline recurrence. Motivated by this observation,
we characterize translation invariant operators that preserve the multivariate B-spline recurrence and analogous results are
also provided for the multivariate cube spline.
Keywords multivariate B-spline recurrence - cube splines - translation invariant operators - homogeneous distributions
Mathematics Subject Classifications (2000) 41A15 - 44A35
Dedicated to Carl de Boor with appreciation for his insight into B-splines.
Charles A. Micchelli was supported in part by the US National Science of Foundation under grant CCR-0407476.
Yuesheng Xu was supported in part by the US National Science Foundation under grant CCR-0407476, by the Natural Science Foundation
of China under grant 10371122, by the Chinese Academy of Sciences under the program “One Hundred Distinguished Chinese Scientists”
and by Ministry of Education, People’s Republic of China, under the Changjian Scholarship through Zhongshan University.