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.