# Regarding the*p*-norms of radial basis interpolation matrices

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DOI: 10.1007/BF01303522

- Cite this article as:
- Baxter, B.J.C., Sivakumar, N. & Ward, J.D. Constr. Approx (1994) 10: 451. doi:10.1007/BF01303522

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## Abstract

A radial basis function approximation has the form where

*ϕ*:*R*^{d}→*R*is some given (usually radially symmetric) function, (*y*_{j})_{1}^{n}are real coefficients, and the centers (*x*_{j})_{1}^{n}are points in*R*^{d}. For a wide class of functions ϕ, it is known that the interpolation matrix*A*=(ϕ(*x*_{j}−*x*_{k}))_{j,k=1}^{n}is invertible. Further, several recent papers have provided upper bounds on ||*A*^{−1}||_{2}, where the points (*x*_{j})_{1}^{n}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}for*p*≥1 which apply when ϕ decays sufficiently quickly and*A*is symmetric and positive definite. We include an application of this analysis to a preconditioning of the interpolation matrix*A*_{n}= (ϕ(*j*−*k*))_{j,k=1}^{n}when ϕ(*x*)=(*x*^{2}+*c*^{2})^{1/2}, the Hardy multiquadric. In particular, we show that sup_{n}||*A*_{n}^{−1}||_{∞}is finite. Furthermore, we find that the bi-infinite symmetric Toeplitz matrix enjoys the remarkable property that ||*E*^{−1}||_{p}= ||*E*^{−1}||_{2}for every*p*≥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.### AMS classification

41A0541A63### Key words and phrases

Positive definiteRadial functionsPolya frequency functionsToeplitz matricesMultiquadrics## Copyright information

© Springer-Verlag New York Inc 1994