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In the present note we consider a type of matrices stemming in the context ofthe numerical approximation of distributed order fractional differentialequations (FDEs): from one side they could look standard, since they are, real,symmetric and positive definite. On the other hand they present specificdifficulties which prevent the successful use of classical tools. In particularthe associated matrix-sequence, with respect to the matrix-size, isill-conditioned and it is such that a generating function does not exists, butwe face the problem of dealing with a sequence of generating functions with anintricate expression. Nevertheless, we obtain a real interval where thesmallest eigenvalue belongs, showing also its asymptotic behavior. We observethat the new bounds improve those already present in the literature and give amore accurate spectral information, which are in fact used in the design offast numerical algorithms for the associated large linear systems,approximating the given distributed order FDEs. Very satisfactory numericalresults are presented and critically discussed, while a section withconclusions and open problems ends the current note.

Fine spectral estimates with applications to the optimally fast solution of large FDE linear systems