Regarding thep-norms of radial basis interpolation matrices

Abstract

A radial basis function approximation has the form

whereϕ:R ^d→R is some given (usually radially symmetric) function, (y _j ) ⁿ₁ are real coefficients, and the centers (x _j ) ⁿ₁ are points inR ^d. For a wide class of functions ϕ, it is known that the interpolation matrixA=(ϕ(x _j −x _k )) ⁿ _j,k=1 is invertible. Further, several recent papers have provided upper bounds on ||A ⁻¹||₂, where the points (x _j ) ⁿ₁ satisfy the condition ||x _j −x _k ||₂≥δ,j≠k, for some positive constant δ. In this paper we calculate similar upper bounds on ||A ⁻¹||₂ forp≥1 which apply when ϕ decays sufficiently quickly andA is symmetric and positive definite. We include an application of this analysis to a preconditioning of the interpolation matrixA _n = (ϕ(j−k)) ⁿ _j,k=1 when ϕ(x)=(x ²+c ²)^1/2, the Hardy multiquadric. In particular, we show that sup _n ||A ⁻¹ _n ||_∞ is finite. Furthermore, we find that the bi-infinite symmetric Toeplitz matrix

enjoys the remarkable property that ||E ⁻¹|| _p = ||E ⁻¹||₂ for everyp≥1 when ϕ is a Gaussian. Indeed, we also show that this property persists for any function ϕ which is a tensor product of even, absolutely integrable Pólya frequency functions.