Asymptotic spectral properties of totally symmetric multilevel Toeplitz matrices

Let n = (n(1),..., n(k)) be a multiindex and kappa(n) = Pi(k)(j=1) n(j). We say that n -> infinity if n(i) -> infinity, 1 <= i <= k. If r = (r(1), -.., r(k)) and s = (s(1),..., s(k)), let vertical bar r - s vertical bar = (vertical bar r(1) - s(1)vertical bar,..., vertical bar s(1) - s(k)vertical bar). We say that a multilevel Toeplitz matrix of the form T-n = [t(vertical bar r-s vertical bar)](r,s=1)(infinity) is totally symmetric. Let Q(k) be the k-fold Cartesian product of Q = [-pi, pi] with itself, and let {t(r)}(r=-infinity)(infinity) be the Fourier coefficients of a function f=f (theta(1),..., theta(k)) in L-2(Q(k)) that is even in each variable theta(1),..., theta(k), so that T-n is totally symmetric for every n. We associate the multiindex n with 2(k) multiindices m(n, p), 0 <= p <= 2(k) - 1, such that lim(n ->infinity)kappa(m(n, p))/kappa(n) = 2(-k), 0 <= p <= 2(k)-1, and Sigma(2k-1)(p=0) kappa(m(n, p) = kappa(n), and show that the singular values of T-n separate naturally into 2(k) sets S-n,0,... S-n, 2(k)-1 with cardinalities kappa(m(n, 0)),..., kappa(m(n, 2(k)-1)) such that the singular values in each set S-n,S-p are associated with singular vectors exhibiting a particular type of symmetry. Our main result is that the singular values in S-n,S-p and the singular values of T-m(n,T- p) are absolutely equally distributed with respect to the class zeta of functions bounded and uniformly continuous on R as n -> infinity, 0 <= p <= 2(k)-1. If f is real-valued, then an analogous result holds for the eigenvalues and eigenvectors of T-n. (c) 2005 Elsevier Inc. All rights reserved.