REGARDING THE P-NORMS OF RADIAL BASIS INTERPOLATION MATRICES
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abstract
A radial basis function approximation has the form where {symbol}:RdR is some given (usually radially symmetric) function, (yj)1n are real coefficients, and the centers (xj)1n are points in Rd. For a wide class of functions {symbol}, it is known that the interpolation matrix A=({symbol}(xj-xk))j,k=1n is invertible. Further, several recent papers have provided upper bounds on ||A-1||2, where the points (xj)1n satisfy the condition ||xj-xk||2, jk, for some positive constant . In this paper we calculate similar upper bounds on ||A-1||2 for p1 which apply when {symbol} decays sufficiently quickly and A is symmetric and positive definite. We include an application of this analysis to a preconditioning of the interpolation matrix An = ({symbol}(j-k))j,k=1n when {symbol}(x)=(x2+c2)1/2, the Hardy multiquadric. In particular, we show that supn ||An-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 p1 when {symbol} is a Gaussian. Indeed, we also show that this property persists for any function {symbol} which is a tensor product of even, absolutely integrable Plya frequency functions. 1994 Springer-Verlag New York Inc.
