We may build polynomials of the form:

[math]y = \sum_{k=0}^K C_{k}t^k [/math]

[math]y\in \mathbb{C} [/math]

where [math]t[/math] is time and [math]C_{k}[/math] is the polynomial’s coefficient vector, by specifying sets of [math]R[/math] point-wise conditions:

[math]y_{D,r}(t_{r}) = \zeta_{r}[/math]

where [math]D,r[/math] is the derivative order, [math]t_{r}[/math] is the point of time observation, and [math]\zeta_{r}[/math] is the value for condition r.

Now we simply move from the point-wise conditioned form to the general polynomial form. For [math]R[/math] specified non-singular conditions we will need a polynomial of at most degree [math]K = R-1[/math]. Each condition can be represented by:

[math]y_{D,r}(t_{r}) = \sum_{k=D,r}^{K}(C_{k} \frac{t_{r}^{k-D,r}}{k+1} \prod_{d=0}^{D,r}(k-d+1)) = \zeta_{r}[/math]

and we extract the desired coefficient vector by organizing all conditions into matrix form and inverting the [math]RxR[/math] condition matrix of constants, [math]G[/math]:

[math]\begin{bmatrix} C_{k} \end{bmatrix} = \begin{bmatrix} G \end{bmatrix}^{-1} \begin{bmatrix} \zeta_{r} \end{bmatrix}[/math]