diagonal dominance is achieved only for B-spline degrees $d = 1, \ldots, 6$ (easily verifiable by a computer). The corresponding Geršgorin bounds are presented in Table \[tbl3.1\]. This directly proves de Boor’s conjecture for low order B-splines.

[cIccccc]{} [ ]{}$n\backslash d$ & 2 & 3 & 4 & 5 & 6\
[ ]{} 64 & 1.998136 & 2.994873 & 4.785918 & 7.466648 & 11.727897\
128 & 1.999541 & 2.998757 & 4.796641 & 7.492176 & 11.785901\
256 & 1.999886 & 2.999694 & 4.799180 & 7.498105 & 11.799106\
512 & 1.999971 & 2.999924 & 4.799797 & 7.499534 & 11.802256\
1024 & 1.999993 & 2.999981 & 4.799950 & 7.499884 & 11.803026\
2048 & 1.999998 & 2.999995 & 4.799987 & 7.499971 & 11.803216\
[ ]{}$\textrm{GB}(d)$ & 2 & 3 & $\frac{96}{19} \approx 5.052632$ & 10 & $\frac{5760}{127} \approx 45.354331$\
[ ]{}

Note that in the case of tridiagonal Töplitz matrices, i.e. for $d = 2, 3$, and, thus, $r = 1$ in (\[3.1\]), the exact eigenvalues are also known (see Böttcher–Grudsky [@Boettcher-Grudsky-2005]) $$\lambda_k (T_n^d) = t_0^d + 2 t_1^d \cos \frac{\pi k}{n - d + 1}, \quad
  d = 2, 3, \quad k = 1, \ldots, n - d.$$ The largest and the smallest 