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Frames for the solution of operator equations in Hilbert spaces with fixed dual pairing

Balazs, Peter and Harbrecht, Helmut. (2019) Frames for the solution of operator equations in Hilbert spaces with fixed dual pairing. Numerical Functional Analysis and Optimization, 40 (1). pp. 65-84.

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Abstract

For the solution of operator equations, Stevenson introduced a definition of frames, where a Hilbert space and its dual are not  identified. This means that the Riesz isomorphism is not used as an identification, which, for example, does not make sense for the Sobolev spaces $H_0^1(Omega)$ and $H^{-1}(Omega)$ . In this article, we are going to revisit the concept of Stevenson frames and introduce it for Banach spaces. This is equivalent to $ell^2$ -Banach frames. It is known that, if such a system exists, by defining a new inner product and using the Riesz isomorphism, the Banach space is isomorphic to a Hilbert space. In this article, we deal with the contrasting setting, where $mathcal{H}$ and $mathcal{H}'$ are not identified, and equivalent norms are distinguished, and show that in this setting the investigation of $ell^2$ -Banach frames make sense.
Faculties and Departments:05 Faculty of Science > Departement Mathematik und Informatik > Mathematik > Computational Mathematics (Harbrecht)
UniBasel Contributors:Harbrecht, Helmut
Item Type:Article, refereed
Article Subtype:Research Article
Publisher:Taylor & Francis
ISSN:0163-0563
e-ISSN:1532-2467
Note:Publication type according to Uni Basel Research Database: Journal article
Language:English
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Last Modified:22 Mar 2019 15:17
Deposited On:22 Mar 2019 15:16

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