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On the Singular Values of Generalized Toeplitz Matrices

Authors:

Bin?Shao mailto:bshao@math.scu.edu" title=" bshao@math.scu.edu" itemprop=" email" data-track=" click" data-track-action=" Email author" data-track-label=" " >Email author

Affiliation:

(1) Department of Mathematics and Computer Science, Santa Clara University, Santa Clara, 95053, CA, USA

Abstract:

For a bounded function sgr

defined on $$ [0, 1]times mathbb{T} $$

, let ${s^{(n)}_{k}}^{n+1}_{k=1}$ be the set of singular values of the (n + 1) x (n + 1) matrix whose (j, k)-entries are equal to

$frac{1}{2pi}int^{2pi}_{0} sigma(frac{k}{n},e^{itheta})e^{-i(j-k)theta}dtheta,qquad j,k = 0, 1,...,n.$

These matrices can be thought of as variable-coefficient Toeplitz matrices orgeneralized Toeplitz matrices. Matrices of the above form can be also thoughtof as the discrete analogue of pseudodifferential operators. Under a certainsmoothness assumption on the function sgr

, we prove that

$sum_{k=1}^{n+1} f(s^{2}_{k})quad = quad c_{1}cdot (n+1) + c_{2} + o(1)qquad as enskip n rightarrow infty,$

where the constant c₁ and a part of c₂ are shown to have explicit integralrepresentations. The other part of c₂ turns out to have a resemblance to theToeplitz case. This asymptotic formula can be viewed as a generalization ofthe classical theory on singular values of Toeplitz matrices.

Keywords:

Mathematics Subject Classification (2000). Primary: 47 Secondary: 46

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