 Cubic cardinal spline interpolation with two left endpoint derivatives

For knot spacing $h=1/2$ on $[0,1]$, the five translates indexed $-3,-2,-1,0,1$ span the $C^2$ piecewise cubic space. With normalized knot values $1/6,2/3,1/6$, nodal interpolation gives $(c_{-3}+4c_{-2}+c_{-1})/6=u(0)$ and the two shifted equations at $h,2h$. Left-end Hermite data give $(c_{-1}-c_{-3})/(2h)=u^{\prime }(0)$ and $(c_{-3}-2c_{-2}+c_{-1})/h^2=u^{\prime \prime }(0)$. These five equations uniquely determine the coefficients. Derivative data need existence or separate prescription; continuity of $u$ alone does not supply them.