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- W1490253183 abstract "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 $pin (1, 2)$, 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 $p > 2$." @default.
- W1490253183 created "2016-06-24" @default.
- W1490253183 creator A5051673818 @default.
- W1490253183 date "2010-06-12" @default.
- W1490253183 modified "2023-09-27" @default.
- W1490253183 title "Conditionally Positive Functions and p-norm Distance Matrices" @default.
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- W1490253183 hasPublicationYear "2010" @default.
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