Frames for the Solution of Operator Equations in Hilbert Spaces with Fixed Dual Pairing. (2nd January 2019)

• Record Type: Journal Article
• Authors: Balazs, Peter; Harbrecht, Helmut
• Is Part Of: Numerical functional analysis and optimization. Volume 40:Number 1(2019)
• Page Start: 65
• Page End: 84
• Publication Date: 2019-01-02
• Subjects: Banach frames -- discretization of operators -- frames -- invertibility -- matrix representation -- Stevenson frames; Functional analysis -- Periodicals; Numerical analysis -- Periodicals; Mathematical optimization -- Periodicals; Numerical Analysis, Computer-Assisted; 515.705
• DOI: 10.1080/01630563.2018.1495232
• Languages: English
• ISSNs: 0163-0563

Abstract

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 spacesH 0 1 ( Ω ) andH − 1 ( Ω ) . In this article, we are going to revisit the concept of Stevenson frames and introduce it for Banach spaces. This is equivalent toℓ 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, whereH andH ′ are not identified, and equivalent norms are distinguished, and show that in this setting the investigation ofℓ 2 -Banach frames make sense.