Multivariate interpolation

Remarks

For $d = 1$ and $V$ the polynomials of degree at most $N - 1$ this is polynomial interpolation. For $d \ge 2$ and $V$ a fixed space of polynomials in several variables with $\dim V = N$, the interpolant is unique exactly when the points are a unisolvent point set 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.