---
tag: C4B5FB6A34
kind: definition
name: Multivariate interpolation
msc: [41A63, 41A05, 65D05]
defines: BM703GV34G
uses: [0JPCHTXWG0, 07GNX8GCYP, 0XPR1E85W0, SS1YMS2NPC]
sources:
  - url: https://en.wikipedia.org/wiki/Multivariate_interpolation
---

## Definition

Let $d \ge 2$, let $\Omega \subseteq \R^d$, let $x_1, \dots, x_N \in \Omega$ be pairwise distinct points of [[Euclidean space|0XPR1E85W0]], and let $y_1, \dots, y_N \in \R$.
Let $V$ be a set of real [[functions of several variables|0JPCHTXWG0]] on $\Omega$.
A *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 $i = 1, \dots, N$.
*Multivariate interpolation* is the problem of finding such an $s$, usually from a space $V$ chosen so that it exists and is unique.

## Remarks

For $d = 1$ and $V$ the polynomials of degree at most $N - 1$ this is [[polynomial interpolation|SS1YMS2NPC]].
For $d \ge 2$ 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 $V$.

Example: $d = 2$, $V = \set{a + bx + cy}$, points $(0, 0)$, $(1, 0)$, $(0, 1)$ with values $1, 3, 4$.
Then $s(x, y) = 1 + 2x + 3y$ is an interpolant: $s(0, 0) = 1$, $s(1, 0) = 3$, $s(0, 1) = 4$.

Non-example: with the same $V$ and the collinear points $(0, 0)$, $(1, 1)$, $(2, 2)$ with values $0, 1, 3$, there is no interpolant.
Indeed $s(t, t) = a + (b + c) t$ is affine in $t$, but the values $0, 1, 3$ at $t = 0, 1, 2$ do not lie on a line.