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  "body": "\n## Definition\n\nLet $d \\ge 2$, let $\\Omega \\subseteq \\R^d$, let $x_1, \\dots, x_N \\in \\Omega$ be pairwise distinct points of [[Euclidean space|0XPR1E85W0]], and let $y_1, \\dots, y_N \\in \\R$.\nLet $V$ be a set of real [[functions of several variables|0JPCHTXWG0]] on $\\Omega$.\nA *multivariate interpolant* of the data $(x_i, y_i)$ from $V$ is a function $s \\in V$ such that $s(x_i) = y_i$ for $i = 1, \\dots, N$.\n*Multivariate interpolation* is the problem of finding such an $s$, usually from a space $V$ chosen so that it exists and is unique.\n\n## Remarks\n\nFor $d = 1$ and $V$ the polynomials of degree at most $N - 1$ this is [[polynomial interpolation|SS1YMS2NPC]].\nFor $d \\ge 2$ and $V$ a fixed space of [[polynomials in several variables|07GNX8GCYP]] with $\\dim V = N$, the interpolant is unique exactly when the points are a [[unisolvent point set|5276MDSDQV]] for $V$.\n\nExample: $d = 2$, $V = \\set{a + bx + cy}$, points $(0, 0)$, $(1, 0)$, $(0, 1)$ with values $1, 3, 4$.\nThen $s(x, y) = 1 + 2x + 3y$ is an interpolant: $s(0, 0) = 1$, $s(1, 0) = 3$, $s(0, 1) = 4$.\n\nNon-example: with the same $V$ and the collinear points $(0, 0)$, $(1, 1)$, $(2, 2)$ with values $0, 1, 3$, there is no interpolant.\nIndeed $s(t, t) = a + (b + c) t$ is affine in $t$, but the values $0, 1, 3$ at $t = 0, 1, 2$ do not lie on a line.\n",
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