In the case of cardinal B-splines, we have already concluded that the splines are symmetric, and therefore the maximum values of $N^d$ is attained at the middle of the support, for $x = (d + 1) /2$. The maximum value is $$N^d\left( \frac{d + 1}{2} \right) = N^d(x_0^{\ast}) = t_0^d.$$ Furthermore, unimodality implies that the values of the spline $N^d$ are decreasing in the interval $[ (d + 1) /2, d + 1 ]$, so $$t_0^d > t_1^d > \cdots > t_r^d.
  \label{3.2}$$

To esitmate the condition number of a cardinal B-spline we need to bound both the minimal and the maximal singular value of $T_n^d$. For a symmetric and positive definite matrix, the singular values are eigenvalues. Therefore, the bounds for the eigenvalues of $T_n^d$ are sought for. From the Geršgorin bound for the eigenvalues, and the partition of unity of the B-spline basis, we obtain an upper bound for $\lambda_{\max}(T_n^d)$ $$\lambda_{\max}(T_n^d) \leq t_0^d + 2 (t_1^d + \cdots + t_r^d) = 1.
  \label{3.3}$$ Similarly, we also obtain a lower bound for $\lambda_{\min}(T_n^d)$, $$\lambda_{\min}(T_n^d) \geq t_0^d - 2 (t_1^d + \cdots + t_r^d) = 2t_0^d - 1,
  \label{3.4}$$ which is sensible only if $T_n^d$ is strictly diagonally dominant. Strict 