Symmetry and inverse-closedness of matrix algebras and functional calculus for infinite matrices

Gröchenig, Karlheinz; Leinert, Michael

doi:10.1090/s0002-9947-06-03841-4

Cited by 100 publications

(92 citation statements)

References 37 publications

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“…So far, we considered operators (infinite matrices) Φ τ δ , produced from a function φ with a certain decay rate (4). It turns out that all these operators belong in the Gröchenig-Shur class A p,u α [20] which contains operators A = {a m,n } m,n∈Z with norm…”

Section: Local Sampling and Approximationmentioning

confidence: 99%

“…For more details about Wiener's lemma for infinite matrices, we refer to [20,26,34,35], and for an excellent overview on Wiener's lemma and its variations, we refer to [22] and references therein. The bound (19) enables us to deduce that the Riesz basis {ψ τ δ n } n∈Z associated to the reconstruction formula ( 15) inherits the decay rate form φ.…”

Section: Now We Havementioning

confidence: 99%

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Sampling and Approximation in Shift Invariant Subspaces of $$L_2(\mathbb {R})$$

Atreas

2020

Springer Optimization and Its Applications

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Let φ be a continuous function in L 2 (R) with a certain decay at infinity and a non-vanishing property in a neighborhood of the origin for the periodization of its Fourier transform φ. Under the above assumptions on φ, we derive uniform and non-uniform sampling expansions in shift invariant spaces V φ ⊂ L 2 (R). We also produce local (finite) sampling formulas, approximating elements of V φ in bounded intervals of R, and we provide estimates for the corresponding approximation error, namely, the truncation error. Our main tools to obtain these results are the finite section method and the Wiener's lemma for operator algebras.N. Atreas ( )

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Section: Local Sampling and Approximationmentioning

confidence: 99%

Section: Now We Havementioning

confidence: 99%

Sampling and Approximation in Shift Invariant Subspaces of $$L_2(\mathbb {R})$$

Atreas

2020

Springer Optimization and Its Applications

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“…First we define the standard classes of matrices with off‐diagonal decay. For more comprehensive treatments see, e.g., , , , , . In the following A is always a matrix over the index set

double-struckZ

with entries

A (k, l), k, l \in Z

.…”

Section: Norm‐controlled Inversion In Matrix Algebras With Off‐ Diagomentioning

confidence: 99%

Norm‐controlled inversion in smooth Banach algebras, II

Gröchenig

Klotz

2013

Mathematische Nachrichten

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We show that smoothness implies norm‐controlled inversion: the smoothness of an element a in a Banach algebra with a one‐parameter automorphism group is preserved under inversion, and the norm of the inverse a−1 is controlled by the smoothness of a and by spectral data. In our context smooth subalgebras are obtained with the classical constructions of approximation theory and resemble spaces of differentiable functions, Besov spaces or Bessel potential spaces. To treat ultra‐smoothness, we resort to Dales‐Davie algebras. Furthermore, based on Baskakov's work, we derive explicit norm control estimates for infinite matrices with polynomial off‐diagonal decay. This is a quantitative version of Jaffard's theorem.

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“…The simplest case of a trivial action leads to the projective tensor product between L 1 (G) and a C * -algebra. Some references are: [20,4,18,23,22,14,15,3,12,13,5,21,11].…”

Section: Introductionmentioning

confidence: 99%