Conditionally Positive Functions and p-norm Distance Matrices
For researchers in radial basis function interpolation, this clarifies the range of p-norms that ensure unique interpolation, extending known results from Euclidean to general p-norms.
This paper proves that for p-norms with p in (1,2], distance matrices from distinct points are invertible, guaranteeing unique radial basis function interpolation, while for p>2, singular matrices exist, showing interpolation is not always uniquely determined.
In Micchelli's paper "Interpolation of scattered data: distance matrices and conditionally positive functions", deep results were obtained concerning the invertibility of matrices arising from radial basis function interpolation. In particular, the Euclidean distance matrix was shown to be invertible for distinct data. In this paper, we investigate the invertibility of distance matrices generated by $p$-norms. In particular, we show that, for any $p\in (1, 2)$, and for distinct points $ x^1, ..., x^n \in {\cal R}^d $, where $n$ and $d$ may be any positive integers, with the proviso that $ n \ge 2$, the matrix $A \in {\cal R}^{n \times n} $ defined by $$ A_{ij} = \Vert x^i - x^j \Vert_p , \hbox{ for } 1 \le i, j \le n, $$ satisfies $$ (-1)^{n-1}\det A > 0 .$$ We also show how to construct, for every $p > 2$, a configuration of distinct points in some ${\cal R}^d$ giving a singular $p$-norm distance matrix. Thus radial basis function interpolation using $p$-norms is uniquely determined by any distinct data for $p \in (1,2]$, but not so for $p > 2$.