Let f be a cubic interpolant. I want to express it as a sum of "basis functions" multipled by interpolation values. That is,
$$ f(x) = \sum\limits_{i=1}^{n-1} S_i(x)f_i $$ where $f_i$ are the interpolation values at the points $x_i$ and $$ \begin{eqnarray} S_i(x)=\begin{cases} \frac{\tilde{S}_i(x)}{f_i}&\text{ for }x_i\le x<x_{i+1}\\ 0&\text{ otherwise } \end{cases} \end{eqnarray} $$ The $\tilde{S}_i(x)$ are the piecewise cubic splines, which satisfy $$ \tilde{S}_i(x_i) = f_i \\ \tilde{S}_i(x_{i+1}) = f_{i+1} \\ \tilde{S}_i^{'}(x_{i+1}) = \tilde{S}_{i+1}^{'}(x_i) \\ \tilde{S}_i^{''}(x_{i+1}) = \tilde{S}_{i+1}^{''}(x_i)$$
Example: For $x \in [x1,x2[$, we have $f(x) = S_0(x)f_0 + 0 + 0 = \tilde{S}_0(x)$ which, I think, is correct.
Per definition of cubic interpolant, I know that f is twice continuously differentiable, but I am not sure if my notation above represents this fact.