Computable Jordan Decomposition of Linear Continuous Functionals on $C[0;1]$
By the Riesz representation theorem using the Riemann-Stieltjes integral, linear continuous functionals on the set of continuous functions from the unit interval into the reals can either be characterized by functions of bounded variation from the unit interval into the reals, or by signed measures...
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| Format: | Article |
| Language: | English |
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Logical Methods in Computer Science e.V.
2014-09-01
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| Series: | Logical Methods in Computer Science |
| Subjects: | |
| Online Access: | https://lmcs.episciences.org/1117/pdf |
| _version_ | 1797268652579356672 |
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| author | Klaus Weihrauch Tahereh Jafarikhah |
| author_facet | Klaus Weihrauch Tahereh Jafarikhah |
| author_sort | Klaus Weihrauch |
| collection | DOAJ |
| description | By the Riesz representation theorem using the Riemann-Stieltjes integral,
linear continuous functionals on the set of continuous functions from the unit
interval into the reals can either be characterized by functions of bounded
variation from the unit interval into the reals, or by signed measures on the
Borel-subsets. Each of these objects has an (even minimal) Jordan decomposition
into non-negative or non-decreasing objects. Using the representation approach
to computable analysis, a computable version of the Riesz representation
theorem has been proved by Jafarikhah, Lu and Weihrauch. In this article we
extend this result. We study the computable relation between three Banach
spaces, the space of linear continuous functionals with operator norm, the
space of (normalized) functions of bounded variation with total variation norm,
and the space of bounded signed Borel measures with variation norm. We
introduce natural representations for defining computability. We prove that the
canonical linear bijections between these spaces and their inverses are
computable. We also prove that Jordan decomposition is computable on each of
these spaces. |
| first_indexed | 2024-04-25T01:35:53Z |
| format | Article |
| id | doaj.art-b1e658a5762840d2a9e275aafd7ac76f |
| institution | Directory Open Access Journal |
| issn | 1860-5974 |
| language | English |
| last_indexed | 2024-04-25T01:35:53Z |
| publishDate | 2014-09-01 |
| publisher | Logical Methods in Computer Science e.V. |
| record_format | Article |
| series | Logical Methods in Computer Science |
| spelling | doaj.art-b1e658a5762840d2a9e275aafd7ac76f2024-03-08T09:37:13ZengLogical Methods in Computer Science e.V.Logical Methods in Computer Science1860-59742014-09-01Volume 10, Issue 310.2168/LMCS-10(3:13)20141117Computable Jordan Decomposition of Linear Continuous Functionals on $C[0;1]$Klaus WeihrauchTahereh JafarikhahBy the Riesz representation theorem using the Riemann-Stieltjes integral, linear continuous functionals on the set of continuous functions from the unit interval into the reals can either be characterized by functions of bounded variation from the unit interval into the reals, or by signed measures on the Borel-subsets. Each of these objects has an (even minimal) Jordan decomposition into non-negative or non-decreasing objects. Using the representation approach to computable analysis, a computable version of the Riesz representation theorem has been proved by Jafarikhah, Lu and Weihrauch. In this article we extend this result. We study the computable relation between three Banach spaces, the space of linear continuous functionals with operator norm, the space of (normalized) functions of bounded variation with total variation norm, and the space of bounded signed Borel measures with variation norm. We introduce natural representations for defining computability. We prove that the canonical linear bijections between these spaces and their inverses are computable. We also prove that Jordan decomposition is computable on each of these spaces.https://lmcs.episciences.org/1117/pdfcomputer science - logic in computer sciencemathematics - functional analysis |
| spellingShingle | Klaus Weihrauch Tahereh Jafarikhah Computable Jordan Decomposition of Linear Continuous Functionals on $C[0;1]$ Logical Methods in Computer Science computer science - logic in computer science mathematics - functional analysis |
| title | Computable Jordan Decomposition of Linear Continuous Functionals on $C[0;1]$ |
| title_full | Computable Jordan Decomposition of Linear Continuous Functionals on $C[0;1]$ |
| title_fullStr | Computable Jordan Decomposition of Linear Continuous Functionals on $C[0;1]$ |
| title_full_unstemmed | Computable Jordan Decomposition of Linear Continuous Functionals on $C[0;1]$ |
| title_short | Computable Jordan Decomposition of Linear Continuous Functionals on $C[0;1]$ |
| title_sort | computable jordan decomposition of linear continuous functionals on c 0 1 |
| topic | computer science - logic in computer science mathematics - functional analysis |
| url | https://lmcs.episciences.org/1117/pdf |
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