Positive weights give a bounded partition-of-unity NURBS basis

Each tensor product $N_{\alpha }$ is nonnegative by [F1]. On any open tensor knot cell, each coordinate lies in one nonempty knot interval, so finite distributivity and [F1] give ${\sum }_{\alpha \in \mathcal{I}}{N}_{\alpha }(\xi )={\prod }_{r=1}^d({\sum }_{i_r}{B}_{i_r,p_r,{\mathrm{\Xi }}_r}^{(r)}({\xi }_r))=1.$ Hence $0\le N_{\alpha }\le 1$ on the cell. The endpoint values follow by the one-sided endpoint convention in B-spline spaces and knot conventions.

Since every coefficient $w_{\alpha }$ is between $w_{-}$ and $w_{+}$, the identity in step 1.1 yields $w_{-}{\sum }_{\alpha }{N}_{\alpha }\le W={\sum }_{\alpha }{w}_{\alpha }{N}_{\alpha }\le w_{+}{\sum }_{\alpha }{N}_{\alpha },$ so $w_{-}\le W\le w_{+}$. Thus each $R_{\alpha }$ is well-defined and nonnegative, and ${\sum }_{\alpha }{R}_{\alpha }=\frac{{\sum }_{\alpha }{w}_{\alpha }{N}_{\alpha }}{W}=1.$ For any coefficients $c_{\alpha }$, ${\sum }_{\alpha }{c}_{\alpha }{R}_{\alpha }=(1\mathrm{/}W){\sum }_{\alpha }(c_{\alpha }{w}_{\alpha })N_{\alpha }$. Multiplication of each coefficient by the nonzero $w_{\alpha }$ is a bijection on the finite coefficient space, so the span equals $S_h\mathrm{/}W$.

On the open knot cell all $N_{\alpha }$ and $W$ are polynomial and $W\ge w_{-}>0$ by step 2.1. The ordinary quotient rule gives the displayed gradient formula. Also $\mathrm{\parallel }\mathrm{\nabla }W{\mathrm{\parallel }}_{\mathrm{\infty }}\le {\sum }_{\alpha }{w}_{\alpha }\mathrm{\parallel }\mathrm{\nabla }{N}_{\alpha }{\mathrm{\parallel }}_{\mathrm{\infty }}\le G_W$ and $N_{\alpha }\le 1$ by step 1.1. Applying these bounds to the quotient rule proves the stated derivative estimate. The finite number of polynomial pieces makes each $G_N$ finite.