On certain (block) Toeplitz matrices related to radial functions

Interpolation of smooth functions and the discretization of elliptic PDEs by means of radial functions lead to structured linear systems which, for equidistant grid points, have almost the (block) Toeplitz structure. We prove upper bounds for the condition numbers of the $n \times n$ $n\times n$Toeplitz matrices which discretize the model problem $u''(x)=f(x)$, $x \in (0,1)$, $u(0)=a$, $u(1)=b$, $u''(x)=f(x)$over an equally spaced grid of $n+2$ $n+2$ points in $[0,1]$ $[0,1]$by means of the collocation method based on radial functions of the multiquadric, inverse multiquadric and Gaussian type. These bounds are asymptotically sharp.