\[tbl5.1\], the smallest eigenvalues of the collocation matrices converge rapidly and monotonically to the smallest eigenvalues of the corresponding circulant periodizations $C_m^d$, as well as to the limiting value (\[5.2\]).

![Spectral condition numbers of Töplitz matrices $T_n^9$ (lower, brighter line), and the circulant periodizations $C_m^9$ (solid black line). The constant function denotes $1/\lambda_{\infty}^9$.[]{data-label="fig5.2"}](k9tc.eps){width="8cm"}

It is worth noting that the spectral bounds obtained in such a way for lower degrees ($d = 2, \ldots, 6$) of cardinal B-splines are quite sharper than those established by the Geršgorin circle theorem (cf. Table \[tbl3.1\] and Table \[tbl5.1\]), at no additional cost.

[cIccIccIcc]{} [ ]{}$n\backslash d$ & $T_n^2$ & $C_m^2$ & $T_n^5$ & $C_m^5$ & $T_n^6$ & $C_m^6$\
[ ]{} 64 & 1.998137 & 1.998758 & 7.466648 & 7.472749 & 11.72790 & 11.74214\
128 & 1.999541 & 1.999694 & 7.492176 & 7.493492 & 11.78590 & 11.78866\
256 & 1.999886 & 1.999924 & 7.498105 & 7.498410 & 11.79911 & 11.79971\
512 & 1.999971 & 1.999981 & 7.499534 & 7.499607 & 11.80226 & 11.80240\
1024 & 1.999993 & 1.999995 & 7.499884 & 7.499902 & 11.80303 & 11.80306\
2048 & 1.999998 & 1.999999 & 7.499971 & 7.499975 & 11.80322 & 11.80322\
$1/\lambda_{\infty}^d$ & & 2.000000 & & 