and we can take $h = 1$. So, just for simplicity, we shall consider only the cardinal B-splines, i.e., B-splines with simple knots placed at successive integers. It should be stressed that the only free parameters in this problem are the *degree* $d$ of the B-splines and the *size* $\nu$ of the interpolation problem. Our aim is to prove that the condition of $A$ can be bounded independently of its order.

The corresponding collocation matrices $A$ are symmetric, positive definite, and most importantly, Töplitz. But, it is not easy to compute the elements of $A^{-1}$, or even reasonably sharp estimates of their magnitudes. So, the natural choice of norm in (\[1.1\]) is the spectral norm $$\kappa_2(A) = \frac{\sigma_{\max}(A)}{\sigma_{\min}(A)},
  \label{1.2}$$ where $\sigma_{\max}(A)$, and $\sigma_{\min}(A)$ denote the largest and the smallest singular value of $A$, respectively.

For low spline degrees $d \leq 6$, the collocation matrices are also strictly diagonally dominant, and it is easy to bound $\kappa_2(A)$ by a constant. Consequently, de Boor’s conjecture is valid for $d \leq 6$.

For higher degrees, the condition number can be estimated by embedding the Töplitz matrix $A$ into circulant matrices of higher orders. The main advantage of this technique, developed by Davis [@DavisP-94] 