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  "body": "\n## Definition\n\nLet $n \\ge 1$, let $a = x_0 \u003c x_1 \u003c \\cdots \u003c x_k = b$ with $k \\ge 1$, and let $y_0, \\dots, y_k \\in \\R$.\nA *spline interpolant of degree $n$* of the data $(x_i, y_i)$ is a [[spline|84TJ8KQ926]] $s$ of degree $n$ with knots $x_0, \\dots, x_k$ such that $s(x_i) = y_i$ for $i = 0, \\dots, k$.\n\n## Remarks\n\nThe splines of degree $n$ with these knots form a space of dimension $n + k$, and the interpolation conditions are $k + 1$ equations.\nSo $n - 1$ further conditions are needed for uniqueness.\nFor $n = 1$ the interpolant is unique: it is the broken line through the data.\nFor $n = 3$ two end conditions are added.\nExamples are $s''(a) = s''(b) = 0$ for the natural [[cubic spline|V42H0R3EKX]], and prescribed values of $s'(a)$ and $s'(b)$ for the clamped cubic spline.\nEach of these choices gives a unique interpolant.\n\nExample: data $(0, 0)$, $(1, 1)$, $(2, 0)$ with $n = 1$.\nThe 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 $1$.\n\nNon-example: for the same data, $s(x) = 2x - x^2$ takes the values $0, 1, 0$.\nBut it is not a spline of degree $1$, because its pieces have degree $2$.\nIt is a [[polynomial interpolant|SS1YMS2NPC]], not a linear spline interpolant.\n",
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