Cubic spline

Remarks

This is the spline of degree $3$. Writing $s_i$ for the piece on $[t_{i-1}, t_i]$, clause 2 says that $s_i(t_i) = s_{i+1}(t_i)$, $s_i'(t_i) = s_{i+1}'(t_i)$ and $s_i''(t_i) = s_{i+1}''(t_i)$ for $1 \le i \le k - 1$, where $'$ is the derivative. The cubic splines with fixed knots form a vector space of dimension $k + 3$. Given values at all knots, there is exactly one natural cubic spline taking them.

Example: knots $-1 < 0 < 1$ and $s(x) = 0$ for $x \le 0$, $s(x) = x^3$ for $x \ge 0$. At $0$ both pieces have value, first derivative and second derivative $0$, so $s$ is a cubic spline. It is not natural, since $s''(1) = 6 \ne 0$.

Non-example: $s(x) = \abs{x}$ on $[-1, 1]$ with the knots $-1 < 0 < 1$ satisfies clause 1, since its pieces $-x$ and $x$ are polynomials. It fails clause 2: its first derivative jumps from $-1$ to $1$ at $0$.