The Generalized Operator Based Prony Method
The generalized Prony method is a reconstruction technique for a large variety of sparse signal models that can be represented as sparse expansions into eigenfunctions of a linear operator A . However, this procedure requires the evaluation of higher powers of the linear operator A that are often ex...
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| Veröffentlicht in: | Constructive approximation 2020-10, Vol.52 (2), p.247-282 |
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| creator | Stampfer, Kilian Plonka, Gerlind |
| description | The generalized Prony method is a reconstruction technique for a large variety of sparse signal models that can be represented as sparse expansions into eigenfunctions of a linear operator
A
. However, this procedure requires the evaluation of higher powers of the linear operator
A
that are often expensive to provide. In this paper we propose two important extensions of the generalized Prony method that essentially simplify the acquisition of the needed samples and, at the same time, can improve the numerical stability of the method. The first extension regards the change of operators from
A
to
φ
(
A
)
, where
φ
is a suitable operator-valued mapping, such that
A
and
φ
(
A
)
possess the same set of eigenfunctions. The goal is now to choose
φ
such that the powers of
φ
(
A
)
are much simpler to evaluate than the powers of
A
. The second extension concerns the choice of the sampling functionals. We show how new sets of different sampling functionals
F
k
can be applied with the goal being to reduce the needed number of powers of the operator
A
(resp.
φ
(
A
)
) in the sampling scheme and to simplify the acquisition process for the recovery method. |
| doi_str_mv | 10.1007/s00365-020-09501-6 |
| format | Article |
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A
. However, this procedure requires the evaluation of higher powers of the linear operator
A
that are often expensive to provide. In this paper we propose two important extensions of the generalized Prony method that essentially simplify the acquisition of the needed samples and, at the same time, can improve the numerical stability of the method. The first extension regards the change of operators from
A
to
φ
(
A
)
, where
φ
is a suitable operator-valued mapping, such that
A
and
φ
(
A
)
possess the same set of eigenfunctions. The goal is now to choose
φ
such that the powers of
φ
(
A
)
are much simpler to evaluate than the powers of
A
. The second extension concerns the choice of the sampling functionals. We show how new sets of different sampling functionals
F
k
can be applied with the goal being to reduce the needed number of powers of the operator
A
(resp.
φ
(
A
)
) in the sampling scheme and to simplify the acquisition process for the recovery method.</description><identifier>ISSN: 0176-4276</identifier><identifier>EISSN: 1432-0940</identifier><identifier>DOI: 10.1007/s00365-020-09501-6</identifier><language>eng</language><publisher>New York: Springer US</publisher><subject>Analysis ; Eigenvectors ; Linear operators ; Mathematics ; Mathematics and Statistics ; Numerical Analysis ; Numerical stability ; Prony's method ; Sampling</subject><ispartof>Constructive approximation, 2020-10, Vol.52 (2), p.247-282</ispartof><rights>The Author(s) 2020</rights><rights>The Author(s) 2020. This work is published under https://creativecommons.org/licenses/by/4.0/ (the “License”). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License.</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c363t-8bb5a7ffba7439f7b1143194b29e1286551b4aee38eb72256a2e59b83a835ac53</citedby><cites>FETCH-LOGICAL-c363t-8bb5a7ffba7439f7b1143194b29e1286551b4aee38eb72256a2e59b83a835ac53</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s00365-020-09501-6$$EPDF$$P50$$Gspringer$$Hfree_for_read</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s00365-020-09501-6$$EHTML$$P50$$Gspringer$$Hfree_for_read</linktohtml><link.rule.ids>314,780,784,27924,27925,41488,42557,51319</link.rule.ids></links><search><creatorcontrib>Stampfer, Kilian</creatorcontrib><creatorcontrib>Plonka, Gerlind</creatorcontrib><title>The Generalized Operator Based Prony Method</title><title>Constructive approximation</title><addtitle>Constr Approx</addtitle><description>The generalized Prony method is a reconstruction technique for a large variety of sparse signal models that can be represented as sparse expansions into eigenfunctions of a linear operator
A
. However, this procedure requires the evaluation of higher powers of the linear operator
A
that are often expensive to provide. In this paper we propose two important extensions of the generalized Prony method that essentially simplify the acquisition of the needed samples and, at the same time, can improve the numerical stability of the method. The first extension regards the change of operators from
A
to
φ
(
A
)
, where
φ
is a suitable operator-valued mapping, such that
A
and
φ
(
A
)
possess the same set of eigenfunctions. The goal is now to choose
φ
such that the powers of
φ
(
A
)
are much simpler to evaluate than the powers of
A
. The second extension concerns the choice of the sampling functionals. We show how new sets of different sampling functionals
F
k
can be applied with the goal being to reduce the needed number of powers of the operator
A
(resp.
φ
(
A
)
) in the sampling scheme and to simplify the acquisition process for the recovery method.</description><subject>Analysis</subject><subject>Eigenvectors</subject><subject>Linear operators</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Numerical Analysis</subject><subject>Numerical stability</subject><subject>Prony's method</subject><subject>Sampling</subject><issn>0176-4276</issn><issn>1432-0940</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2020</creationdate><recordtype>article</recordtype><sourceid>C6C</sourceid><recordid>eNp9kE9LAzEQxYMoWKtfwNOCR4lO_idHLbYKlXqo55C0s7al7tZke6if3ugK3jzNPHjvzfAj5JLBDQMwtxlAaEWBAwWngFF9RAZMCl6khGMyAGY0ldzoU3KW8waAKSvMgFzPV1hNsMEUtutPXFazXVm7NlX3IRf5ktrmUD1jt2qX5-SkDtuMF79zSF7HD_PRI53OJk-juyldCC06amNUwdR1DEYKV5vIyiPMycgdMm61UizKgCgsRsO50oGjctGKYIUKCyWG5Krv3aX2Y4-585t2n5py0nNphHEgnC0u3rsWqc05Ye13af0e0sEz8N9QfA_FFyj-B4rXJST6UC7m5g3TX_U_qS_On2Jc</recordid><startdate>20201001</startdate><enddate>20201001</enddate><creator>Stampfer, Kilian</creator><creator>Plonka, Gerlind</creator><general>Springer US</general><general>Springer Nature B.V</general><scope>C6C</scope><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>20201001</creationdate><title>The Generalized Operator Based Prony Method</title><author>Stampfer, Kilian ; Plonka, Gerlind</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c363t-8bb5a7ffba7439f7b1143194b29e1286551b4aee38eb72256a2e59b83a835ac53</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2020</creationdate><topic>Analysis</topic><topic>Eigenvectors</topic><topic>Linear operators</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Numerical Analysis</topic><topic>Numerical stability</topic><topic>Prony's method</topic><topic>Sampling</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Stampfer, Kilian</creatorcontrib><creatorcontrib>Plonka, Gerlind</creatorcontrib><collection>Springer Nature OA/Free Journals</collection><collection>CrossRef</collection><jtitle>Constructive approximation</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Stampfer, Kilian</au><au>Plonka, Gerlind</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>The Generalized Operator Based Prony Method</atitle><jtitle>Constructive approximation</jtitle><stitle>Constr Approx</stitle><date>2020-10-01</date><risdate>2020</risdate><volume>52</volume><issue>2</issue><spage>247</spage><epage>282</epage><pages>247-282</pages><issn>0176-4276</issn><eissn>1432-0940</eissn><abstract>The generalized Prony method is a reconstruction technique for a large variety of sparse signal models that can be represented as sparse expansions into eigenfunctions of a linear operator
A
. However, this procedure requires the evaluation of higher powers of the linear operator
A
that are often expensive to provide. In this paper we propose two important extensions of the generalized Prony method that essentially simplify the acquisition of the needed samples and, at the same time, can improve the numerical stability of the method. The first extension regards the change of operators from
A
to
φ
(
A
)
, where
φ
is a suitable operator-valued mapping, such that
A
and
φ
(
A
)
possess the same set of eigenfunctions. The goal is now to choose
φ
such that the powers of
φ
(
A
)
are much simpler to evaluate than the powers of
A
. The second extension concerns the choice of the sampling functionals. We show how new sets of different sampling functionals
F
k
can be applied with the goal being to reduce the needed number of powers of the operator
A
(resp.
φ
(
A
)
) in the sampling scheme and to simplify the acquisition process for the recovery method.</abstract><cop>New York</cop><pub>Springer US</pub><doi>10.1007/s00365-020-09501-6</doi><tpages>36</tpages><oa>free_for_read</oa></addata></record> |
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| ispartof | Constructive approximation, 2020-10, Vol.52 (2), p.247-282 |
| issn | 0176-4276 1432-0940 |
| language | eng |
| recordid | cdi_proquest_journals_2473790398 |
| source | SpringerNature Journals |
| subjects | Analysis Eigenvectors Linear operators Mathematics Mathematics and Statistics Numerical Analysis Numerical stability Prony's method Sampling |
| title | The Generalized Operator Based Prony Method |
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