Norm Estimates for Inverses of Toeplitz Distance Matrices Original Research Article

A radial basis function approximation has the form [formula] where φ: [0,∞)→ R is some given function, (yj)n1 are real coefficients, and the centres (xj)n1 are points in Rd. For a wide class of functions φ, it is known that the interpolation matrix A = (φ(||xj − xk||2))nj,k=1 is invertible. Further, several recent papers have provided upper bounds on ||A−1||2, where the points (xj)n1 satisfy the condition ||xj − xk||2 ≥ δ, j ≠ k, for some positive constant δ. In this paper, we provide the least upper bound on ||A−1||2 when the points (xj)n1 form any subset of the integer lattice Ld, and when φ is a conditionally negative definite function of order 1, a large set of functions which includes the multiquadric. Specifically, for any set of points (xj)n1 ⊂ Ld, we provide the inequality [formula] where e = [1, . . . , 1]T ∈ Rd and where φ̂ is the generalized Fourier transform of φ. 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,