Asymptotic spectral properties of totally symmetric multilevel Toeplitz matrices

Let n = (n1 , … , nk) be a multiindex and . We say that n → ∞ if ni → ∞, 1 ⩽ i ⩽ k. If r = (r1, … , rk) and s = (s1, … , sk), let ∣r − s∣ = (∣r1 − s1∣, … , ∣s1 − sk∣). We say that a multilevel Toeplitz matrix of the form is totally symmetric. Let Qk be the k-fold Cartesian product of Q = [−π, π] with itself, and let be the Fourier coefficients of a function f = f(θ1, … , θk) in L2(Qk) that is even in each variable θ1, … , θk, so that Tn is totally symmetric for every n. We associate the multiindex n with 2k multiindices m(n, p), 0 ⩽ p ⩽ 2k − 1, such that limn→∞κ(m (n, p))/κ(n) = 2−k, 0 ⩽ p ⩽ 2k − 1, and , and show that the singular values of Tn separate naturally into 2k sets with cardinalities κ(m(n, 0)), … , κ(m(n, 2k − 1)) such that the singular values in each set are associated with singular vectors exhibiting a particular type of symmetry. Our main result is that the singular values in and the singular values of Tm(n, p) are absolutely equally distributed with respect to the class G of functions bounded and uniformly continuous on R as n → ∞, 0 ⩽ p ⩽ 2k − 1. If f is real-valued, then an analogous result holds for the eigenvalues and eigenvectors of Tn.