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The analysis of the spectral features of a Toeplitz matrix-sequence$\left\{T_{n}(f)\right\}_{n\in\mathbb N}$, generated by a symbol $f\inL^1([-\pi,\pi])$, real-valued almost everywhere (a.e.), has been provided ingreat detail in the last century, as well as the study of the conditioning,when $f$ is nonnegative a.e. Here we consider a novel type of problem arisingin the numerical approximation of distributed-order fractional differentialequations (FDEs), where the matrices under consideration take the form \[\mathcal{T}_{n}=c_0T_{n}(f_0)+c_{1} h^h T_{n}(f_{1})+c_{2} h^{2h}T_{n}(f_{2})+\cdots+c_{n-1} h^{(n-1)h}T_{n}(f_{n-1}), \] $c_0,c_{1},\ldots,c_{n-1} \in [c_*,c^*]$, $c^*\ge c_*>0$, independent of $n$, $h=\frac{1}{n}$,$f_j\sim g_j$, $g_j=|\theta|^{2-jh}$, $j=0,\ldots,n-1$. Since the resultinggenerating function depends on $n$, the standard theory cannot be applied andthe analysis has to be performed using new ideas. Few selected numericalexperiments are presented, also in connection with matrices that come fromdistributed-order FDE problems, and the adherence with the theoretical analysisis discussed together with open questions and future investigations.

On the extreme eigenvalues and asymptotic conditioning of a class of Toeplitz matrix-sequences arising from fractional problems