I am having some trouble understanding what the question below is asking. What does the given polynomial $P(x)$ have to do with deriving the not-a-knot spline interpolant for $S(x)$? Also, since not-a-knot is a boundary condition, what does it mean to derived it for $S(x)$?
For general data points $(x_1, y_1), (x_2, y_2),...,(x_N , y_N )$, where $x_1 < x_2 < . .. < x_N$ and $N \geq 4$, Assume that S(x) is a cubic spline interpolant for four data points $(x_1, y_1)$, $(x_2, y_2)$, $(x_3, y_3)$, and $(x_4, y_4)$ $ S(x) = \begin{cases} p_1(x), & [x_1,x_2] \\ p_2(x), & [x_2,x_3] \\ p_3(x), & [x_3,x_4] \\ \end{cases} $ Suppose $P (x) = 2x^3 + 5x +7$ is the cubic interpolant for the same four points $(x_1, y_1)$, $(x_2, y_2)$, $(x_3, y_3)$, $(x_4, y_4)$ where $x_1 < x_2 < x_3 < x_4$ are knots. What is the not-a-knot spline interpolant $S(x)$?