The Singularity of Distance Matrices

Abstract

There has been a lot of interest recently in the problem of interpolation of data defined on ℝⁿ by radial basis functions. In its most elementary form the problem is as follows. Suppose x ₁,...,x _m are given points in ℝⁿ, and ║ · ║ is any norm on ℝⁿ. Then a linear subspace of C(ℝⁿ) is defined as the span of the functions h _i ,: ℝⁿ → ℝ where h _i (x) = ║x − x _i ║, 1 ≤ i≤ m. The interpolation problem is then to determine conditions on the points xi such that for any m real numbers d ₁ ,..., d _m there exist unique constants c ₁,...,c _m such that

$$\sum\limits_{i=1}^{m}{{{c}_{i}}}\left\| {{x}_{i}} \right.-\left. {{x}_{i}} \right\|={{d}_{j}},j=1,2,\ldots ,m.$$

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