2018/10/08 by Ferrari, Paola, Furci, Isabella, Hon, Sean +2 · 2 citations
#15B05 #65F08 #65F15 #FOS: Mathematics #Numerical Analysis (math.NA)
paper · doi:10.48550/arxiv.1810.03326
Given a Lebesgue integrable function f over [0,2π], we consider the sequence of matrices \YnTn[f]\n, where Tn[f] is the n-by-n Toeplitz matrix generated by f and Yn is the flip permutation matrix, also called the anti-identity matrix. Because of the unitary character of Yn, the singular values of Tn[f] and Yn Tn[f] coincide. However, the eigenvalues are affected substantially by the action of the matrix Yn. Under the assumption that the Fourier coefficients are real, we prove that \YnTn[f]\n is distributed in the eigenvalue sense as ϕg(θ)=\ g(θ), · amp; θ∈ [0,2π], -g(-θ), · amp; θ∈ [-2π,0), . with g(θ)=|f(θ)|. We also consider the preconditioning introduced by Pestana and Wathen and, by using the same arguments, we prove that the preconditioned sequence is distributed in the eigenvalue sense as ϕ1, under the mild assumption that f is sparsely vanishing. We emphasize that the mathematical tools introduced in this setting have a general character and in fact can be potentially used in different contexts. A number of numerical experiments are provided and critically discussed.