Asymptotic spectra of large matrices coming from the symmetrization of Toeplitz structure functions and applications to preconditioning

The singular value distribution of the matrix-sequence {YnTn[f]}n, with Tn[f] generated by (Formula presented.), was shown in [J. Pestana and A.J. Wathen, SIAM J Matrix Anal Appl. 2015;36(1):273-288]. The results on the spectral distribution of {YnTn[f]}n were obtained independently in [M. Mazza and J. Pestana, BIT, 59(2):463-482, 2019] and [P. Ferrari, I. Furci, S. Hon, M.A. Mursaleen, and S. Serra-Capizzano, SIAM J. Matrix Anal. Appl., 40(3):1066-1086, 2019]. In the latter reference, the authors prove that {YnTn[f]}n is distributed in the eigenvalue sense as (Formula presented.) under the assumptions that f belongs to (Formula presented.) and has real Fourier coefficients. The purpose of this paper is to extend the latter result to matrix-sequences of the form {h(Tn[f])}n, where h is an analytic function. In particular, we provide the singular value distribution of the sequence {h(Tn[f])}n, the eigenvalue distribution of the sequence {Ynh(Tn[f])}n, and the conditions on f and h for these distributions to hold. Finally, the implications of our findings are discussed, in terms of preconditioning and of fast solution methods for the related linear systems.