Norm estimates for inverses of Toeplitz distance matrices

j=1 yj φ(‖x − xj‖2), x ∈ R , where φ: [0,∞) → R is some given function, (yj) n 1 are real coefficients, and the centres (xj) n 1 are points in R. For a wide class of functions φ, it is known that the interpolation matrix A = (φ(‖xj − xk‖2)) n j,k=1 is invertible. Further, several recent papers have provided upper bounds on ‖A‖2, where the points (xj) n 1 satisfy the condition ‖xj − xk‖2 ≥ δ, j 6= k, for some positive constant δ. In this paper, we provide the least upper bound on ‖A‖2 when the points (xj) n 1 form any subset of the integer lattice Z, 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) n 1 ⊂ Z , we provide the inequality ‖A‖2 ≤ ( ∑