Index Calculus for Approximation Methods and Singular Value Decomposition

Roch, Steffen; Silbermann, Bernd

doi:10.1006/jmaa.1998.6022

Cited by 20 publications

(16 citation statements)

References 6 publications

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“…The so-called splitting phenomenon, which was discovered by Roch and Silbermann [11,12] and is also discussed in detail in Section 4.3 of [6], reveals that the minimal singular value s (n) 1 of T n (a) may be equal to zero or converge to zero even if ess inf(|a|) > 0. This happens for instance if a : T → C\{0} is continuous and has nonzero winding number about the origin.…”

Section: Introduction and Main Resultsmentioning

confidence: 93%

Maximum norm versions of the Szegő and Avram–Parter theorems for Toeplitz matrices

Bogoya

Böttcher

Grudsky

et al. 2015

Journal of Approximation Theory

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The collective behavior of the singular values of large Toeplitz matrices is described by the Avram-Parter theorem. In the case of Hermitian matrices, the Avram-Parter theorem is equivalent to Szegő's theorem on the eigenvalues. The Avram-Parter theorem in conjunction with an improvement made by Trench implies estimates in the mean between the singular values and the appropriately ordered absolute values of the symbol. The purpose of this paper is twofold. Under natural hypotheses, we first strengthen the known estimates in the mean to estimates in the maximum norm, thus turning from collective results on the singular values to results on individual singular values. Secondly, we want to emphasize that the use of the quantile function eases the proofs and statements of results significantly and provides a promising language for forthcoming research into higher order asymptotics for individual singular values. (J.M. Bogoya), aboettch@mathematik.tu-chemnitz.de (A. Böttcher), grudsky@math.cinvestav.mx (S.M. Grudsky), maximenko@esfm.ipn.mx (E.A. Maximenko).

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Section: Introduction and Main Resultsmentioning

confidence: 93%

Maximum norm versions of the Szegő and Avram–Parter theorems for Toeplitz matrices

Bogoya

Böttcher

Grudsky

et al. 2015

Journal of Approximation Theory

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“…The k-splitting property of approximation numbers for {T n (a)} with k > 0 was studied only since the 1990s. For p = 2 Roch and one of the authors [9,10] have proved the following result: if a Toeplitz operator T (a) with a piecewise continuous generating function a is Fredholm (i.e. invertible modulo compact operators) on 2 , then the singular values of T n (a) have the k-splitting property with…”

Section: T N (A) Is Stable On P ⇐⇒ W 1 T N (A) = T (A) and W 2 T N (Amentioning

confidence: 95%

Approximation numbers for the finite sections of Toeplitz operators with piecewise continuous symbols

Rogozhin

Silbermann

2006

Journal of Functional Analysis

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In this paper we discuss the asymptotic distribution of the approximation numbers of the finite sections for a Toeplitz operator T (a) ∈ L( p ), 1 < p < ∞, where a is a piecewise continuous function on the unit circle. We prove that the behavior of the approximation numbers of the finite sections T n (a) = P n T (a)P n depends heavily on the Fredholm properties of the operators T (a) and T (ã) (ã(t) = a(1/t)). In particular, if the operators T (a) and T (ã) are Fredholm on p , then the approximation numbers of T n (a) have the so-called k-splitting property. But, in contrast with the case of continuous symbols, the splitting number k is in general larger than dim ker T (a) + dim ker T (ã).

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“…In the works of S. Roch and one of the authors (see [6,9]) this theorem has already been proved for p = 2 and µ = 0, that is, for the case where {s (2) k (T n (a))} nN k =1 are the singular values of T n (a). The asymptotic behavior of the approximation numbers of the matrices T n (a) for 1 < p < ∞ was treated by A. Böttcher (see [1,3]).…”

Section: Introductionmentioning

confidence: 91%

“…Moreover, the search for an estimate was one of the reasons to study the asymptotic distribution of the approximation numbers in the more general situation at hand. The investigations carried out in [6] and [9] are heavily based on C*-algebra techniques, which provide only the existence of the k-splitting phenomenon. (However, in Bötcher's paper [1] an estimate for the convergence of the kth approximation number to zero has already been given for N = 1.)…”

Section: Introductionmentioning

confidence: 99%