Natural cubic spline interpolant
= Natural cubic spline interpolant
{title2=$N\mathbf v,\qquad(N\mathbf v)(x_i)=v_i$}
For at least two distinct knots $x_1<\cdots<x_n$ and values $v_i$, the unique <natural cubic spline> matching those values and linear beyond $x_1,x_n$. Its interpolation map is a <linear map> by uniqueness. Over a larger domain $[a,b]$, the two exterior pieces remain linear. With only one knot, arbitrary slopes of <affine functions> make uniqueness false.