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- W2070263263 abstract "Let n = (n1 , … , nk) be a multiindex and κ(n̲)=∏j=1knj. 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 Tn̲=[t|r̲-s̲|]r̲,s̲=1̲∞̲ is totally symmetric. Let Qk be the k-fold Cartesian product of Q = [−π, π] with itself, and let {tr̲}r̲=-∞̲∞̲ 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 ∑p=02k-1κ(m̲(n̲,p)=κ(n̲), and show that the singular values of Tn separate naturally into 2k sets Sn̲,0,…,Sn̲,2k-1 with cardinalities κ(m(n, 0)), … , κ(m(n, 2k − 1)) such that the singular values in each set Sn̲,p are associated with singular vectors exhibiting a particular type of symmetry. Our main result is that the singular values in Sn̲,p 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." @default.
- W2070263263 created "2016-06-24" @default.
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- W2070263263 date "2006-07-01" @default.
- W2070263263 modified "2023-09-29" @default.
- W2070263263 title "Asymptotic spectral properties of totally symmetric multilevel Toeplitz matrices" @default.
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- W2070263263 doi "https://doi.org/10.1016/j.laa.2005.12.012" @default.
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