Spline

Remarks

Splines of degree $0$ are usually taken to be step functions, constant on each $[t_{i-1}, t_i)$, with jumps allowed at the knots. For $n = 1$, clause 2 asks that $s$ is continuous, so $s$ is a broken line. The splines of degree $n$ with fixed knots form a vector space of dimension $n + k$. Some authors allow lower smoothness at some knots; this is encoded by repeating knots.

Example: with knots $-1 < 0 < 1$ and $n = 3$, the function $s(x) = 0$ for $x \le 0$ and $s(x) = x^3$ for $x \ge 0$ is a cubic spline. Each piece is a polynomial of degree at most $3$. At $0$ the values, first derivatives and second derivatives of both pieces are $0$, so $s$ is twice continuously differentiable.

Non-example: with the same knots, $s(x) = 0$ for $x \le 0$ and $s(x) = x^2$ for $x \ge 0$ is not a cubic spline. Its second derivative jumps from $0$ to $2$ at $0$, so clause 2 fails.