I'm in a Numerical Analysis class and I'm working on a term project that talks about how we can derive multivariate interpolating polynomials from a dataset of bivariate data (for some real-valued function f(x, y) -> z)
I know that for the univariate case, the unique Lagrange interpolation polynomial is:
$p(x)=\sum_{i=0}^{n}{y_il_i(x)}$
$l_i(x)=\prod_{j=0,j\neq{i}}^{n}{\frac{x-x_j}{x_i-x_j}}$
Where i = 0, .., n for n + 1 interpolation nodes, and $l_i$ represents the cardinal polynomial for the $i$th node. In this respect, the $y_i$ terms act as scaling factors and the $l_i$ acts as a continuous analog to the binary Kronecker delta (Cheney & Kincaid, 2008, p. 126-127).
In order to find an analog to the multivariate case, I discovered the following definition of the Lagrange form in a 2008 research article by Gasca et al (url below):
$p(x)=\sum_{i=0}^{n}{y_i\prod_{j=0,j\neq{i}}^{n}\frac{H_j(x)}{H_j(x_i)}}$
Where for each point $x_i$, a hyperplane $H_i$ (with a corresponding affine function) containing $x_i$ but no other interpolation nodes $x_j$ is chosen. This geometric interpretation of the Kronecker delta allows us to expand the Lagrange polynomial to a higher dimensional space.
I don't know anything about hyperplanes, but I get how as we add more dimensions to our problem space we will end up going from points to lines to planes to higher dimensional structures. I don't see where Gasca define the uniqueness of this multivariate Lagrange poly, however. This makes me a little concerned.
Would anyone be able to help me with this? Are we able to say that the interpolating polynomial of this form is unique for any give multivariate datasets? What would be a counterexample if this is not the case?
Any help or feedback is greatly appreciated.
Primary Reference: https://link.springer.com/article/10.1023/A:1018981505752