B-spline spaces and knot conventions

Let $p\in {\mathbb{N}}_0$, $n\ge p+1$, and $a<b$. A clamped knot vector is a nondecreasing finite sequence ${\xi }_1=\cdots ={\xi }_{p+1}=a<{\xi }_{p+2}\le \cdots \le {\xi }_n<{\xi }_{n+1}=\cdots ={\xi }_{n+p+1}=b,$ with ${\xi }_j<{\xi }_{j+p+1}$ for every $1\le j\le n$. Empty interior ranges are omitted. The multiplicity $\mu (\eta )$ of an interior knot $\eta$ is its number of occurrences among ${\xi }_{p+2},\dots ,{\xi }_n$.

Write ${\mathcal{P}}_p$ for the real polynomials of degree at most $p$. The degree-$p$ spline space $S_{p,\xi }$ consists of functions that restrict to a member of ${\mathcal{P}}_p$ on each nonempty knot interval and whose derivatives through order $p-\mu (\eta )$ agree across each interior knot $\eta$. The convention $C^{-1}$ means that no matching condition is imposed. For this piecewise definition, take knot cells left-closed and right-open; at a knot with no matching condition, the value is the one from the cell on its right, and at $b$ use the left limit. Thus the function value at a discontinuous knot is specified.

For each degree $d\in \{0,\dots ,p\}$, the functions $B_{j,d,\xi }$ are indexed by $1\le j\le n+p-d$. Set $B_{j,0,\xi }(x)={\mathbf{1}}_{[{\xi }_j,{\xi }_{j+1})}(x),1\le j\le n+p\mathrm{.}$ For $1\le d\le p$ and $1\le j\le n+p-d$, define recursively $B_{j,d,\xi }(x)=\frac{x-{\xi }_j}{{\xi }_{j+d}-{\xi }_j}{B}_{j,d-1,\xi }(x)+\frac{{\xi }_{j+d+1}-x}{{\xi }_{j+d+1}-{\xi }_{j+1}}{B}_{j+1,d-1,\xi }(x),$ where a term with zero denominator is defined to be zero. For every degree $d$ and every allowed index $j$, set $B_{j,d,\xi }(b)$ equal to its left limit. In particular, the degree-$p$ family has indices $1\le j\le n$. Its basis and regularity properties are proved in B-spline basis properties and polynomial reproduction ↗.