The Szegő and Avram–Parter theorems for general test functions

Böttcher, Albrecht; Grudsky, Sergei M.; Maksimenko, E. A.

doi:10.1016/j.crma.2008.06.002

Cited by 5 publications

(6 citation statements)

References 6 publications

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“…In summary, from (51) we have T * n,α T n,α = T µα ( |f | (2) ) + R n,m,α + N n,m,α , with {T µα ( |f | (2) )} n ∼ σ ( |f | (2) , Q). We recall that, owing to (55), the relation |f…”

Section: Singular Value Distribution For the Sequence {[T N Z Nα |0]} Nmentioning

confidence: 97%

“…-if k is not a multiple of α, then e In summary, from (51) we have (2) , Q). We recall that, owing to (55), the relation |f (2) , Q).…”

Section: Some Remarks Are In Ordermentioning

confidence: 98%

“…(of relation (55).) We denote by a j the Fourier coefficients of |f | (2) . We want to show that for r, c = 0, .…”

Section: Singular Value Distribution For the Sequence {[T N Z Nα |0]} Nmentioning

confidence: 99%

See 2 more Smart Citations

Spectral Features and Asymptotic Properties for g-Circulants and g-Toeplitz Sequences

Ngondiep

Serra‐Capizzano²,

Sesana³

2010

SIAM J. Matrix Anal. & Appl.

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For a given nonnegative integer α, a matrix A n of size n is called α-Toeplitz if its entries obey the rule A n = [a r−αs ] n−1 r,s=0 . Analogously, a matrix A n again of size n is called α-circulant if A n = a (r−αs) mod n n−1 r,s=0 . Such kind of matrices arises in wavelet analysis, subdivision algorithms and more generally when dealing with multigrid/multilevel methods for structured matrices and approximations of boundary value problems. In this paper we study the singular values of α-circulants and we provide an asymptotic analysis of the distribution results for the singular values of α-Toeplitz sequences in the case where {a k } can be interpreted as the sequence of Fourier coefficients of an integrable function f over the domain (−π, π). Some generalizations to the block, multilevel case, amounting to choose f multivariate and matrix valued, are briefly considered.Theorem 2.1. Let B be an arbitrary (complex) m × n matrix. Then: (a) There exists a unitary m × m matrix U and a unitary n × n matrix V such that U * BV = Σ is an m × n "diagonal matrix" of the following form: Σ = D 0 0 0 , D := diag(σ 1 , . . . , σ r ), σ 1 ≥ σ 2 ≥ · · · ≥ σ r > 0.Here σ 1 , . . . , σ r are the nonvanishing singular values of B, and r is the rank of B.

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Section: Singular Value Distribution For the Sequence {[T N Z Nα |0]} Nmentioning

confidence: 97%

“…-if k is not a multiple of α, then e In summary, from (51) we have (2) , Q). We recall that, owing to (55), the relation |f (2) , Q).…”

Section: Some Remarks Are In Ordermentioning

confidence: 98%

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Spectral Features and Asymptotic Properties for g-Circulants and g-Toeplitz Sequences

Ngondiep

Serra‐Capizzano²,

Sesana³

2010

SIAM J. Matrix Anal. & Appl.

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“…The case where the compact set E is a union of m disjoint closed intervals (possibly, degenerate) has been treated in [12] (Theorem 3.6). The general case follows since, for the notion of strong clustering we have to consider the ǫ fattening of E, or D(E, ǫ) defined as in relation (6). It is clear that for every compact set E, the closure of D(E, ǫ) is a finite union of closed intervals and so the general case is reduced to that handled in [12].…”

Section: Eigenvalue Distribution and Clusteringmentioning

confidence: 99%

“…(The book [8] gives a synopsis of all these results in Chapters 5 and 6 and other interesting facts in Chapter 3 concerning the relation between the pseudospectrum of {T n (f )} ∞ n=1 and that of T (f )). The relation (1) was established for a more general class of test functions F in [35,26,6] and the case of functions f of several variables (the multilevel case) and matrix-valued functions was studied in [35] and in [23] in the context of preconditioning (other related results were established by Linnik, Widom, Doktorski, see Section 6.9 in [8]).…”

Section: Introduction and Basic Notationsmentioning

confidence: 99%

The eigenvalue distribution of products of Toeplitz matrices – Clustering and attraction

Serra‐Capizzano¹,

Sesana²,

Strouse³

2010

Linear Algebra and its Applications

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We use a recent result concerning the eigenvalues of a generic (non Hermitian) complex perturbation of a bounded Hermitian sequence of matrices to prove that the asymptotic spectrum of the product of Toeplitz sequences, whose symbols have a real-valued essentially bounded product h, is described by the function h in the "Szegö way". Then, using Mergelyan's theorem, we extend the result to the more general case where h belongs to the Tilli class. The same technique gives us the analogous result for sequences belonging to the algebra generated by Toeplitz sequences, if the symbols associated with the sequences are bounded and the global symbol h belongs to the Tilli class. A generalization to the case of multilevel matrix-valued symbols and a study of the case of Laurent polynomials not necessarily belonging to the Tilli class are also given.

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The part of my path I walked together with Sergei Grudsky

Böttcher

2016

Bol. Soc. Mat. Mex.

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The Szegő and Avram–Parter theorems for general test functions

Cited by 5 publications

References 6 publications

Spectral Features and Asymptotic Properties for g-Circulants and g-Toeplitz Sequences

Spectral Features and Asymptotic Properties for g-Circulants and g-Toeplitz Sequences

The eigenvalue distribution of products of Toeplitz matrices – Clustering and attraction

The part of my path I walked together with Sergei Grudsky

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