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 {Y_nT_n[f]}_n , with T_n[f] generated by $f\in L^1([?\pi ,\pi ])$, 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 {Y_nT_n[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 {Y_nT_n[f]}_n is distributed in the eigenvalue sense as under the assumptions that f belongs to $L^1([?\pi ,\pi ])$ and has real Fourier coefficients. The purpose of this paper is to extend the latter result to matrix‐sequences of the form {h(T_n[f])}_n , where h is an analytic function. In particular, we provide the singular value distribution of the sequence {h(T_n[f])}_n , the eigenvalue distribution of the sequence {Y_nh(T_n[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.