Norm-controlled inversion in smooth Banach algebras, I

Gröchenig, Karlheinz; Klotz, Andreas

doi:10.1112/jlms/jdt004

Cited by 20 publications

(32 citation statements)

References 30 publications

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“…So all three blocks of L have the appropriate decay. Therefore, [27], we know that there is some γ B ∈ ℓ 1 v such that the entries of B 22 and B −1 22 are bounded by γ B for all n. Then, by (17), we have…”

Section: Proofmentioning

confidence: 97%

Localization of Matrix Factorizations

2014

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Matrices with off-diagonal decay appear in a variety of fields in mathematics and in numerous applications, such as signal processing, statistics, communications engineering, condensed matter physics, and quantum chemistry. Numerical algorithms dealing with such matrices often take advantage (implicitly or explicitly) of the empirical observation that this off-diagonal decay property seems to be preserved when computing various useful matrix factorizations, such as the Cholesky factorization or the QRfactorization. There is a fairly extensive theory describing when the inverse of a matrix inherits the localization properties of the original matrix. Yet, except for the special case of band matrices, surprisingly very little theory exists that would establish similar results for matrix factorizations. We will derive a comprehensive framework to rigorously answer the question when and under which conditions the matrix factors inherit the localization of the original matrix for such fundamental matrix factorizations as the LU-, QR-, Cholesky, and Polar factorization.

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Section: Proofmentioning

confidence: 97%

Localization of Matrix Factorizations

2014

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“…As in [9], we can further work with (2.3) to obtain a simpler norm-controlling function and study its asymptotic behavior. and let γ = log 2 (1 + θ).…”

Section: The Last Infinite Product Is Convergent If and Only Ifmentioning

confidence: 99%

“…A admits universal norm-controlled inversion in the sense of Nikolski. It is proved in [9] that if B is a C * -algebra and A is a differential *-subalgebra of B, then A admits norm-controlled inversion in B.…”

Section: Introductionmentioning

confidence: 99%

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Norm-Controlled Inversion in Weighted Convolution Algebras

Samei

Shepelska

2019

J Fourier Anal Appl

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Let G be a discrete group, let p ≥ 1, and let ω be a weight on G. Using the approach from [9], we provide sufficient conditions on a weight ω for ℓ p (G, ω) to be a Banach algebra admitting a norm-controlled inversion in the reduced C * -algebra of G, namely C * r (G). We show that our results can be applied to various cases including locally finite groups as well as finitely generated groups of polynomial or intermediate growth and a natural class of weights on them. These weights are of the form of polynomial or certain subexponential functions. We also consider the non-discrete case and study the existence of norm-controlled inversion in B(L 2 (G)) for some related convolution algebras.

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“…Unlike the inverse-closedness, there are not many papers devoted to the above quantitative version of inverse-closedness [23,24,50,64]. The algebra W of commutative infinite matrices of the form A := (a(i − j)) i,j∈Z with norm A W = j∈Z |a(j)| is inverse-closed in B(ℓ 2 ) by the classical Wiener's lemma [68] but it does not admit norm control [50].…”

Section: Nonlinear Wiener's Lemmamentioning

confidence: 99%

Localized nonlinear functional equations and two sampling problems in signal processing

Sun

2013

Adv Comput Math

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Let 1 ≤ p ≤ ∞. In this paper, we consider solving a nonlinear functional equation f (x) = y, where x, y belong to ℓ p and f has continuous bounded gradient in an inverseclosed subalgebra of B(ℓ 2 ), the Banach algebra of all bounded linear operators on the Hilbert space ℓ 2 . We introduce strict monotonicity property for functions f on Banach spaces ℓ p so that the above nonlinear functional equation is solvable and the solution x depends continuously on the given data y in ℓ p . We show that the Van-Cittert iteration converges in ℓ p with exponential rate and hence it could be used to locate the true solution of the above nonlinear functional equation. We apply the above theory to handle two problems in signal processing: nonlinear sampling termed with instantaneous companding and subsequently average sampling; and local identification of innovation positions and qualification of amplitudes of signals with finite rate of innovation.

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Norm-controlled inversion in smooth Banach algebras, I

Cited by 20 publications

References 30 publications

Localization of Matrix Factorizations

Localization of Matrix Factorizations

Norm-Controlled Inversion in Weighted Convolution Algebras

Localized nonlinear functional equations and two sampling problems in signal processing

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