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Multivariate polynomial interpolation: history

%GascaSauer00b (\JCAM; 122; 2000; 23--35) is a recent and thorough paper on the history on multivariate polynomial interpolation.

I am using this page to hold interesting facts as I come across them and before I can place them more properly. %

%Radon48 \refJ Radon, J.; Zur mechanischen Kubatur; Monatshefte der Math.\ Physik; 52(4); 1948; 286--300; %

First paper to propose use of orthogonal polynomials in the construction of cubature rules with a small number of points for a given degree of exactness. Only one case is worked out, but in explicit detail, namely that of a seven-point rule exact for polynomials of degree $\le 5$. In the process, Radon observes the following: if $T \subset \RR^2$ is correct for $\Pi_k$, and $U$ is a set of $k+2$ points on an arbitrary straight line not meeting $T$, then $T\cup U$ is correct for $\Pi_{k+1}$. Radon makes use of this observation to build up pointsets correct for $\Pi_m$ degree by degree. %

In the Chinese literature, this observation is ascribed to a paper by Liang, X. Z. which appeared in 1965 in a Chinese journal. That paper also has the corresponding result for an arbitrary subset $U$ of cardinality $\dim\Pi_{k+l} - \dim \Pi_k$ on the zeroset of some polynomial $q$ of exact degree $l$ which does not vanish on $T$ giving a pointset $T\cup U$ correct for $\Pi_{k+l}$, at least when $l=2$. However, the cases $l>1$ are essentially different in that there are no obvious facts about the vanishing of $p\in\Pi_{k+l}$ implying that $q\mid p$ nor about pointsets $U$ in the zero set of such $q$ so that vanishing of $p$ on $U$ implies that $p$ vanishes on the entire zeroset of $q$. So, offhand, the conclusion holds if the ideal generated by $q$ is prime (something obviously true when $p$ is a linear polynomial). %

L\"u Chun-Mei has written a thesis in 1997 that explores such questions.