Frames for the Solution of Operator Equations in Hilbert Spaces with Fixed Dual Pairing
| Autor: | Peter Balazs, Helmut Harbrecht |
|---|---|
| Rok vydání: | 2018 |
| Předmět: |
42C15
65J10 47A50 Pure mathematics Control and Optimization Banach space discretization of operators 01 natural sciences symbols.namesake FOS: Mathematics Stevenson frames 0101 mathematics matrix representation Mathematics Mathematics::Functional Analysis Operator (physics) 010102 general mathematics Hilbert space invertibility 16. Peace & justice Functional Analysis (math.FA) Banach frames Computer Science Applications Dual (category theory) Mathematics - Functional Analysis 010101 applied mathematics Sobolev space Pairing Product (mathematics) frames Signal Processing symbols Original Article Isomorphism Analysis |
| Zdroj: | Numerical Functional Analysis and Optimization |
| ISSN: | 1532-2467 0163-0563 |
| Popis: | For the solution of operator equations, Stevenson introduced a definition of frames, where a Hilbert space and its dual are {\em 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. Comment: 23 pages; accepted for publication in 'Numerical Functional Analysis and Optimization' |
| Databáze: | OpenAIRE |
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