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 {T-n(f)}(n is an element of N), generated by the function f is an element of L-1([-pi, pi), 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 formT-n = c(0)T(n)(f(0)) + c(1)h(h)T(n)(f(1)) + c(2)h(2h)T(n)(f(2)) + ... + c(n-1)h((n-1)h)T(n)(f(n-1)),C-0, C-1, . . ., Cn-1 belong to the interval [c(*), c*] with c* >= c(*) > 0 independent of n, h = 1/n, f(j) similar to g(j), and g(j)(theta) = vertical bar theta vertical bar(2-jh) for every j = 0, . . . , n - 1. For nonnegative functions or sequences, the notation s(x) similar to t(x) means that there exist positive constants c, d, independent of the variable x in the definition domain such that cs(x) <= t(x) <= ds(x) for any x. Since the resulting generating func- tion 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.