- 1}]$, $c^T = [t_{n - 1}, \ldots, t_1]$. If $T$ is positive definite, and $|b^T T^{-1} c| < 1$, then $C$ is positive semidefinite.

Once again, there is no obvious efficient way to verify whether the condition $|b^T T^{-1} c| < 1$ is fullfiled or not.

Conjecture about the minimal eigenvalues
========================================

Extensive numerical testing has been conducted, by using *Mathematica* 7 from Wolfram Research, for the symbolic, arbitrary-precision rational, and machine-precision floating-point computations. Whenever feasible, the full accuracy was maintained. Owing mostly to the elegance and the accuracy of these results, insight into and the following conjecture about the spectral properties of the collocation matrices and the corresponding periodizations were obtained.

The circulant $C_m^d$ from (\[4.1\]) is always positive definite, and the index $\mu$ of its smallest eigenvalue $\lambda_\mu(C_m^d)$ is always the integer nearest to $m / 2$, i.e., $$\lambda_{\mu}(C_m^d) =
  \begin{cases}
    \displaystyle\lambda_{\frac{m \pm 1}{2}}(C_m^d)
    = t_0^d + 2 \sum_{j=1}^r(-1)^j t_j^d \cos\left(\frac{\pi j}{m}\right), &
    \hbox{$m$ odd,} \\
    \displaystyle\lambda_{\frac{m}{2}}(C_m^d) = t_0^d + 2 \sum_{j=1}^r(-1)^j t_j^d, &
    \hbox{$m$ even.}
  \end{cases}
  \label{5.1}$$ \[con5.1\]

Figure \[fig5.1\] illustrates both cases of Conjecture \[con5.1\].

![The eigenvalues (black dots) $\lambda_k(C_m^d)$ for spline of degree $d = 7$ with $n = 23, 24$, respectively. The associated 