Direct and Inverse Theorems in the Theory of Approximation of Banach Space Vectors by Exponential Type Entire Vectors

Grushka, Yaroslav; Torba, Sergiy

doi:10.1007/978-3-7643-9919-1_17

Cited by 2 publications

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“…In particular, we need a theory of smoothness based on the action of an automorphism group (Section 3), and a spectral characterization of bandlimited elements (Section 5.1). Related results were obtained independently in [30,47].…”

Section: Characterization Of Smoothness By Approximationsupporting

confidence: 60%

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Noncommutative Approximation: Inverse-Closed Subalgebras and Off-Diagonal Decay of Matrices

2010

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We investigate two systematic constructions of inverse-closed subalgebras of a given Banach algebra or operator algebra A, both of which are inspired by classical principles of approximation theory. The first construction requires a closed derivation or a commutative automorphism group on A and yields a family of smooth inverse-closed subalgebras of A that resemble the usual Hölder-Zygmund spaces. The second construction starts with a graded sequence of subspaces of A and yields a class of inverse-closed subalgebras that resemble the classical approximation spaces. We prove a theorem of Jackson-Bernstein type to show that in certain cases both constructions are equivalent.These results about abstract Banach algebras are applied to algebras of infinite matrices with off-diagonal decay. In particular, we obtain new and unexpected conditions of off-diagonal decay that are preserved under matrix inversion.

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Section: Characterization Of Smoothness By Approximationsupporting

confidence: 60%

“…This is all we need, but clearly the abstract theory works for much more general group actions [30,47].…”

Section: Remark the Inverse-closedness Ofmentioning

confidence: 99%