The BSpline routine computes a piecewise function representing the B-spline of order k in the symbol v.  The non-zero portions of this function are polynomials of degree k-1. If the knots option is not provided, then the uniform knot list $\[0,1,...,k\]$ is used.

The knot list must contain exactly k+1 elements.  These elements must be in non-decreasing order; otherwise, unexpected results may be produced.  The knots can have a multiplicity up to k-1.  If the multiplicity of a knot is m, then the continuity at that knot is $C\left(k-m-1\right)$.

This procedure returns a B-spline basis function. Use the CurveFitting[BSplineCurve] procedure to create a B-spline curve.

This function is part of the CurveFitting package, and so it can be used in the form BSpline(..) only after executing the command with(CurveFitting).  However, it can always be accessed through the long form of the command by using CurveFitting[BSpline](..).

$\mathrm{with}\left(\mathrm{CurveFitting}\right)\:$

$\mathrm{BSpline}\left(2\,u\right)$

$\left\{\begin{array}{cc}0& u<0\\ u& u<1\\ 2-u& u<2\\ 0& 2\le u\end{array}\right.$

$\mathrm{BSpline}\left(2\&comma;u\&comma;\mathrm{knots}=\left[0\&comma;a\&comma;2\right]\right)$

$\left\{\begin{array}{cc}0& u<0\\ \frac{u}{a}& u<a\\ \frac{-u+a}{2-a}+1& u<2\\ 0& 2\le u\end{array}\right.$