Spline interpolation

Remarks

The splines of degree $n$ with these knots form a space of dimension $n + k$, and the interpolation conditions are $k + 1$ equations. So $n - 1$ further conditions are needed for uniqueness. For $n = 1$ the interpolant is unique: it is the broken line through the data. For $n = 3$ two end conditions are added. Examples are $s''(a) = s''(b) = 0$ for the natural cubic spline, and prescribed values of $s'(a)$ and $s'(b)$ for the clamped cubic spline. Each of these choices gives a unique interpolant.

Example: data $(0, 0)$, $(1, 1)$, $(2, 0)$ with $n = 1$. The broken line $s(x) = x$ on $[0, 1]$ and $s(x) = 2 - x$ on $[1, 2]$ is continuous, linear on each piece and takes the values $0, 1, 0$, so it is the spline interpolant of degree $1$.

Non-example: for the same data, $s(x) = 2x - x^2$ takes the values $0, 1, 0$. But it is not a spline of degree $1$, because its pieces have degree $2$. It is a polynomial interpolant, not a linear spline interpolant.