A note on the rectangular Schoenberg-Whitney theorem for clamped B-splines
Abstract
Wang et al.(Computer-Aided Design 40 (2008) 999-1008) proposed a knot-subsequence characterization of full row rank for rectangular B-spline collocation matrices. We identify an incorrect inequality in their Appendix proof, which invalidates the argument as written, and make explicit the endpoint and multiplicity conventions on which the criterion depends. The defect lies in the proof, not in the underlying result, which is valid under suitably stated hypotheses; in the lofting construction of Wang et al. these hypotheses are implicitly satisfied, so the lofted curves are unaffected. We state the corrected result and give a new proof, independent of the disputed argument, for clamped B-splines of arbitrary degree with continuous interior multiplicities. In our proof, sufficiency follows from inclusion of spline spaces and knot insertion; necessity selects independent columns and builds the reduced knot vector from them. The proof uses no recursive column-selection construction.
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BibTeX
@misc{tang2026note,
title = {{A note on the rectangular Schoenberg-Whitney theorem for clamped B-splines}},
author = {Shutao Tang and Zixuan Zhao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.24425},
eprint = {2610.24425},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.24425},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A note on the rectangular Schoenberg-Whitney theorem for clamped B-splines
AU - Shutao Tang
AU - Zixuan Zhao
PY - 2026
DA - 2026/10/17
PB - arXiv
DO - 10.48550/arxiv.2610.24425
UR - https://arxiv.org/abs/2610.24425
AB - Wang et al.(Computer-Aided Design 40 (2008) 999-1008) proposed a knot-subsequence characterization of full row rank for rectangular B-spline collocation matrices. We identify an incorrect inequality in their Appendix proof, which invalidates the argument as written, and make explicit the endpoint and multiplicity conventions on which the criterion depends. The defect lies in the proof, not in the underlying result, which is valid under suitably stated hypotheses; in the lofting construction of Wang et al. these hypotheses are implicitly satisfied, so the lofted curves are unaffected. We state the corrected result and give a new proof, independent of the disputed argument, for clamped B-splines of arbitrary degree with continuous interior multiplicities. In our proof, sufficiency follows from inclusion of spline spaces and knot insertion; necessity selects independent columns and builds the reduced knot vector from them. The proof uses no recursive column-selection construction.
ER -