References regarding unisolvent sets

Let $X = {x_1, ..., x_N}$ be a finite subset of $R^n$ and let $p$ and $q$ be any polynomials of degree $k$ or less. X is called $\underline{P_k-unisolvent}$ if $p(x_j) = q(x_j)$ ($j = 1, ..., N$) implies that $p=q$; i.e. evaluation of a polynomial on $X$ uniquely determines that polynomial.

In one dimension, unisolvent sets are completely trivial. But for $n \geq 2$ even confirming whether a given set is unisolvent or not becomes complicated.

The Padua points were recently constructed - a unisolvent set in $R^2$ with minimal growth of their corresponding Lebesgue constant. I am interested in the construction of similar sets in higher dimensions, and have found numerous articles on this particular subset of $R^2$.

I am also interested in $L^p$-norm estimates and interpolant error bounds to Sobolev functions defined on manifolds and more exotic domains. To this end I have seen seen a result used (Duchon, J. 1978) stating that the collection of all $P_k$-unisolvent point sets is open in $R^n$.

I am hoping to find other instances of the use of unisolvent sets, and information on their properties or characterization.

Can anyone provide some references (both introductory and advanced) containing information about unisolvent sets?

1 Answer

The term "unisolvent" is inspired by the much more classical definition involving functions. (e.g. Philip Davis - Interpolation and Approximation, and results by B. Polster) A unisolvent family of functions contains a unique solution to the interpolation problem given a collection of points; Whereas a unisolvent point set will determine a unique solution to the interpolation problem given a particular family of functions.

Books on finite element methods sometimes contain definitions and examples of unisolvent point sets. For example:

Ciarlet, P., G., and J. L. Lions, Handbook of Numerical Analysis: Finite Element Methods

Brenner and Scott, The Mathematical Theory of Finite Element Methods

There are some approximation theory results of L. Bos relating specifically to unisolvent points.