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  "body": "\n## Definition\n\nLet $a \u003c b$, let $k \\ge 1$, and let $a = t_0 \u003c t_1 \u003c \\cdots \u003c t_k = b$ be points called *knots*.\nLet $n \\ge 1$ be an integer.\nA *spline of degree $n$* with these knots is a function $s \\colon [a, b] \\to \\R$ such that:\n\n1. for each $i = 1, \\dots, k$, the restriction of $s$ to $[t_{i-1}, t_i]$ is a [[polynomial function|1G0B73WPX8]] of degree at most $n$;\n2. $s$ is $n - 1$ times continuously differentiable on $[a, b]$.\n\n## Remarks\n\nSplines of degree $0$ are usually taken to be step functions, constant on each $[t_{i-1}, t_i)$, with jumps allowed at the knots.\nFor $n = 1$, clause 2 asks that $s$ is [[continuous|0X3WGX95FE]], so $s$ is a broken line.\nThe splines of degree $n$ with fixed knots form a vector space of dimension $n + k$.\nSome authors allow lower smoothness at some knots; this is encoded by repeating knots.\n\nExample: with knots $-1 \u003c 0 \u003c 1$ 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.\nEach piece is a polynomial of degree at most $3$.\nAt $0$ the values, first [[derivatives|0V1EJCT8G6]] and second derivatives of both pieces are $0$, so $s$ is twice continuously differentiable.\n\nNon-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.\nIts second derivative jumps from $0$ to $2$ at $0$, so clause 2 fails.\n",
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