& t_r^d  &        &        \! \\
      \!        &        &       &               &       &        &        & \vdots & \ddots &        \! \\
      \!        &        &       &               &       &        &        & t_1^d  &        & t_r^d  \! \\
      \hdashline
      \! t_r^d  &        &       &               & t_r^d & \cdots & t_1^d  & t_0^d  & \ddots & \vdots \! \\
      \! \vdots & \ddots &       &               &       & \ddots &        & \ddots & \ddots & t_1^d  \! \\
      \! t_1^d  & \cdots & t_r^d &               &       &        & t_r^d  & \cdots & t_1^d  & t_0^d  \!
    \end{array} \right], \quad
  m = n - d + r.
  \label{4.1}$$ It is obviously a Töplitz matrix with the following symbol $$t = (t_0^d, \ldots, t_r^d, 0, \ldots, 0, t_r^d, \ldots, t_1^d),
  \quad t \in \mathbb{R}^{n - d + r}.$$ This circulant $C_m^d$ is called a periodization of $T_n^d$ by Böttcher and Grudsky [@Boettcher-Grudsky-2005].

The bounds (\[3.3\])–(\[3.4\]) for the eigenvalues of $T_n^d$ are also valid for $C_m^d$. Moreover, $C_m^d$ is doubly stochastic, always having $\lambda_{\max}(C_m^d) = 1$ as its largest eigenvalue. Interestingly enough, the upper bound (\[3.3\]) is attained here (the Geršgorin bounds are rarely so sharp).

The symmetry of $T_n^d$ immediately implies the symmetry of $C_m^d$, and we can 