circulants have order $m = n - 7 + 3$, i.e., $19$ and $20$. Note that for $m = 20$ there is only one minimal eigenvalue, while for $m = 19$ we have two minimal eigenvalues.[]{data-label="fig5.1"}](CS7-23.eps "fig:"){width="5.75cm"} ![The eigenvalues (black dots) $\lambda_k(C_m^d)$ for spline of degree $d = 7$ with $n = 23, 24$, respectively. The associated circulants have order $m = n - 7 + 3$, i.e., $19$ and $20$. Note that for $m = 20$ there is only one minimal eigenvalue, while for $m = 19$ we have two minimal eigenvalues.[]{data-label="fig5.1"}](CS7-24.eps "fig:"){width="5.75cm"}

For even $m$, $\lambda_{\mu}(C_m^d)$ (and, therefore, $\kappa_2(C_m^d)$) depends solely on $d$, i.e., the order $m$ of a circulant is irrelevant here. Moreover, for $m$ odd and even alike, the limiting value of $\lambda_{\mu}(C_m^d)$ is the same: $$\lambda_{\infty}^d := \lim_{m\to\infty} \lambda_{\mu}(C_m^d)
  = t_0^d + 2 \sum_{j=1}^r (-1)^j t_j^d.
  \label{5.2}$$ Hence, the notation $\lambda_{\infty}^d$ is justified, since that value is determined uniquely by the degree $d$ of the chosen cardinal splines. This is consistent with de Boor’s conjecture.

The equations (\[5.1\]) and (\[5.2\]) provide us with efficiently and exactly computable estimates of the spectral condition numbers of large collocation matrices $T_n^d$. As demonstrated in Figure \[fig5.2\] and Table 