I have a sequence of $N$ equispaced knots defined in the interval $[t_0,t_f]$ and a number of values, $x(t_i), i=1...N+1$, that requires perfect matching at these knots. The resulting interpolating function has to assure $C^2$ continuity for $[t_0,t_f]$.
I have solved the above problem using piecewise splines at each interval as $x(t)=\sum^{m}_{j=0}\alpha_jt^j, \> \> t \in [(i-1),i](t_f-t_0)/N$ and considering null first derivatives at $t_0$ and $t_f$ to close the system (values at knots+$C^2$ continuity) considering $m=3$.
I have been looking at B-splines and they have interesting properties since they match $C^2$ continuity intrinsecally (if I am not wrong). However I do not know if they assure perfect interpolation for my values at the knots. Can my problem be solved with B-splines formulation? How?