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

The analysis of the spectral features of a Toeplitz matrix-sequence (Formula presented.), generated by the function (Formula presented.), real-valued almost everywhere (a.e.), has been provided in great 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 arising in the numerical approximation of distributed-order fractional differential equations (FDEs), where the matrices under consideration take the form (Formula presented.) (Formula presented.) belong to the interval (Formula presented.) with (Formula presented.) independent of n, (Formula presented.), (Formula presented.), and (Formula presented.) for every (Formula presented.). For nonnegative functions or sequences, the notation (Formula presented.) means that there exist positive constants c, d, independent of the variable x in the definition domain such that (Formula presented.) for any x. Since the resulting generating function depends on n, the standard theory cannot be applied and the analysis has to be performed using new ideas. Few selected numerical experiments are presented, also in connection with matrices that come from distributed-order FDE problems, and the adherence with the theoretical analysis is discussed, together with open questions and future investigations.