NORM ESTIMATES FOR INVERSES OF TOEPLITZ DISTANCE MATRICES

A radial basis function approximation has the form [GRAPHICS] where phi: [0,infinity)-->R is some given function, (Y-j)(l)(n) are rear coefficients, and the centres (x(j))(1)(n) are points in R(d). For a wide class of functions phi, it is known that the interpolation matrix A = (phi parallel to x(j) - X(k) parallel to 2))(j,k-1)(n) is invertible. Further, several recent papers have provided upper bounds on parallel to A(-1)parallel to(2), where the points (x(j))(1)(n) satisfy the condition parallel to x(j) - X(k) parallel to(2) greater than or equal to delta, j not equal k, for some positive constant delta. In this paper, we provide the least upper bound on parallel to A(-1)parallel to(2) whenthe points (x(j))(1)(n) form any subset of the integer lattice L(d), and when phi is a conditionally negative definite function of order 1, a large set of functions which includes the multiquadric. Specifically, for any set of points (x(j))(1)(n) subset of L(d), we provide the inequality \\A(-1)\\(2)(Sigma/kappa is an element of L(d)\phi(\\pi e+2 pi kappa\\(2))\)(-1), where e = [1,..., 1](T) is an element of R(d) and where phi is the generalized Fourier transform of phi. We provide a constructive proof that no smaller bound is valid and comment on the relevance of the method of analysis to the problem of estimating all the eigenvalues of such an interpolation matrix, (C) 1994 Academic Press, Inc.