eigenvalue can then be uniformly bounded by $$\begin{aligned}
  \lambda_{\max} (T_n^d) & = t_0^d + 2 t_1^d \cos \frac{\pi}{n - d + 1}
    < t_0^d + 2 t_1^d, \\
  \lambda_{\min} (T_n^d) & = t_0^d - 2 t_1^d \cos \frac{\pi}{n - d + 1}
    > t_0^d - 2 t_1^d > 0,\end{aligned}$$ These uniform bounds are somewhat better than those obtained by the Geršgorin circles.

Embeddings of Töplitz matrices into circulants
==============================================

When the degree of a cardinal B-spline is at least $7$, the eigenvalue bounds for Töplitz matrices can be computed by circulant embeddings. First, we will introduce the smallest possible circulant embedding, and give its properties. Then we will present some other known embeddings, with positive semidefinite circulants.

To obtain a bound for $\lambda_{\min}(T_n^d)$, the collocation matrix $T_n^d$ is to be embedded into a circulant $$C_m^d = \left[ \begin{array}{ccccccc:ccc}
      \!        &        &       &               &       &        &        & t_r^d  & \cdots & t_1^d  \! \\
      \!        &        &       &               &       &        &        &        & \ddots & \vdots \! \\
      \!        &        &       &               &       &        &        &        &        & t_r^d  \! \\
      \!        &        &       & \smash{T_n^d} &       &        &        &        &        &        \! \\
      \!        &        &       &               &       &        &        