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  "body": "\n## Definition\n\nLet $m \\ge 0$, let $x_0, \\dots, x_m \\in \\R$ be pairwise distinct, and let $k_0, \\dots, k_m \\ge 1$ be integers with $N = k_0 + \\cdots + k_m$.\nFor each $i$ and each $0 \\le j \\le k_i - 1$ let a real number $y_i^{(j)}$ be given.\nThe *Hermite interpolation polynomial* of these data is the polynomial $p$ with real coefficients such that:\n\n1. $p = 0$ or $\\deg p \\le N - 1$;\n2. $p^{(j)}(x_i) = y_i^{(j)}$ for all $0 \\le i \\le m$ and $0 \\le j \\le k_i - 1$, where $p^{(j)}$ is the $j$-th [[derivative|0V1EJCT8G6]].\n\n## Remarks\n\nSuch a $p$ exists and is unique.\nUniqueness: the difference $q$ of two solutions is $0$ or has [[degree|0P4JRMSWNH]] at most $N - 1$.\nIt has a root of [[multiplicity|047SX3TD69]] at least $k_i$ at each $x_i$, so at least $N$ roots counted with multiplicity, and hence $q = 0$.\nExistence follows because the conditions form a square linear system whose only solution for zero data is $0$.\nIf all $k_i = 1$ this is [[polynomial interpolation|SS1YMS2NPC]]; if $m = 0$ the solution is the Taylor polynomial of order $k_0 - 1$ at $x_0$.\n\nExample: $m = 1$, nodes $0$ and $1$, $k_0 = k_1 = 2$, data $p(0) = 0$, $p'(0) = 0$, $p(1) = 1$, $p'(1) = 0$.\nThen $N = 4$ and $p(x) = 3x^2 - 2x^3$: indeed $p(0) = 0$, $p'(x) = 6x - 6x^2$ gives $p'(0) = p'(1) = 0$, and $p(1) = 1$.\n\nNon-example: $3x^2 - 2x^3 + x^2 (x - 1)^2$ satisfies clause 2 for the same data, since the added term has double roots at $0$ and $1$, but its degree is $4 \u003e 3$, so clause 1 fails.\n\nThe conditions must cover consecutive derivative orders $0, 1, \\dots, k_i - 1$ at each node; dropping this gives [[Birkhoff interpolation|FT1YKVBFC9]], where uniqueness can fail.\n",
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