Natural cubic cardinal spline endpoint rows ($c_{-1}-2c_0+c_1=c_{m-1}-2c_m+c_{m+1}=0$)

For the centered Cardinal cubic B-spline $\beta$ with knot values $1/6,4/6,1/6$, write $s(x)={\sum }_{j=-1}^{m+1}{c}_j\beta (x/h-j)$ on $[0,mh]$. interpolation gives $c_{i-1}+4c_i+c_{i+1}=6u(ih)$ for $i=0,\dots ,m$. Since ${\beta }^{\prime \prime }(0)=-2$ and ${\beta }^{\prime \prime }(\pm 1)=1$, the natural endpoint conditions $s^{\prime \prime }(0)=s^{\prime \prime }(mh)=0$ give the displayed extra two rows. These supply the two equations missing from interpolation alone.