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Pluripotential Numerics
We study the numerical approximation of the fundamental quantities in pluripotential theory , namely the Siciak Zaharjuta extremal plurisubharmonic function V E ∗ of a compact L -regular set E ⊂ C n , its transfinite diameter δ ( E ) , and the pluripotential equilibrium measure μ E : = dd c V E ∗ n...
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| Published in: | Constructive approximation 2019-04, Vol.49 (2), p.227-263 |
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| Format: | Article |
| Language: | English |
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| cited_by | cdi_FETCH-LOGICAL-c316t-378828ab609799fe2564615cc3ae4ba99bed7420d7ab2363cefe4a7be96471ae3 |
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| cites | cdi_FETCH-LOGICAL-c316t-378828ab609799fe2564615cc3ae4ba99bed7420d7ab2363cefe4a7be96471ae3 |
| container_end_page | 263 |
| container_issue | 2 |
| container_start_page | 227 |
| container_title | Constructive approximation |
| container_volume | 49 |
| creator | Piazzon, Federico |
| description | We study the numerical approximation of the fundamental quantities in
pluripotential theory
, namely the
Siciak Zaharjuta extremal plurisubharmonic function
V
E
∗
of a compact
L
-regular set
E
⊂
C
n
, its
transfinite diameter
δ
(
E
)
,
and the
pluripotential equilibrium measure
μ
E
:
=
dd
c
V
E
∗
n
. The developed methods rely on the computation of a
polynomial mesh
for
E
, for which a suitable orthonormal polynomial basis can be defined. We prove the convergence of the approximation of
δ
(
E
)
and the local uniform convergence of our approximation to
V
E
∗
on
C
n
. Then the convergence of the proposed approximation of
μ
E
follows. Our algorithms are based on the properties of polynomial meshes and Bernstein–Markov measures. Numerical tests on some simple cases with
E
⊂
R
2
show the performance of the proposed methods. |
| doi_str_mv | 10.1007/s00365-018-9441-7 |
| format | article |
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pluripotential theory
, namely the
Siciak Zaharjuta extremal plurisubharmonic function
V
E
∗
of a compact
L
-regular set
E
⊂
C
n
, its
transfinite diameter
δ
(
E
)
,
and the
pluripotential equilibrium measure
μ
E
:
=
dd
c
V
E
∗
n
. The developed methods rely on the computation of a
polynomial mesh
for
E
, for which a suitable orthonormal polynomial basis can be defined. We prove the convergence of the approximation of
δ
(
E
)
and the local uniform convergence of our approximation to
V
E
∗
on
C
n
. Then the convergence of the proposed approximation of
μ
E
follows. Our algorithms are based on the properties of polynomial meshes and Bernstein–Markov measures. Numerical tests on some simple cases with
E
⊂
R
2
show the performance of the proposed methods.</description><identifier>ISSN: 0176-4276</identifier><identifier>EISSN: 1432-0940</identifier><identifier>DOI: 10.1007/s00365-018-9441-7</identifier><language>eng</language><publisher>New York: Springer US</publisher><subject>Acquisitions & mergers ; Algorithms ; Analysis ; Approximation ; Convergence ; Markov processes ; Mathematical analysis ; Mathematics ; Mathematics and Statistics ; Numerical Analysis ; Polynomials</subject><ispartof>Constructive approximation, 2019-04, Vol.49 (2), p.227-263</ispartof><rights>Springer Science+Business Media, LLC, part of Springer Nature 2018</rights><rights>Copyright Springer Nature B.V. 2019</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c316t-378828ab609799fe2564615cc3ae4ba99bed7420d7ab2363cefe4a7be96471ae3</citedby><cites>FETCH-LOGICAL-c316t-378828ab609799fe2564615cc3ae4ba99bed7420d7ab2363cefe4a7be96471ae3</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>314,776,780,27901,27902</link.rule.ids></links><search><creatorcontrib>Piazzon, Federico</creatorcontrib><title>Pluripotential Numerics</title><title>Constructive approximation</title><addtitle>Constr Approx</addtitle><description>We study the numerical approximation of the fundamental quantities in
pluripotential theory
, namely the
Siciak Zaharjuta extremal plurisubharmonic function
V
E
∗
of a compact
L
-regular set
E
⊂
C
n
, its
transfinite diameter
δ
(
E
)
,
and the
pluripotential equilibrium measure
μ
E
:
=
dd
c
V
E
∗
n
. The developed methods rely on the computation of a
polynomial mesh
for
E
, for which a suitable orthonormal polynomial basis can be defined. We prove the convergence of the approximation of
δ
(
E
)
and the local uniform convergence of our approximation to
V
E
∗
on
C
n
. Then the convergence of the proposed approximation of
μ
E
follows. Our algorithms are based on the properties of polynomial meshes and Bernstein–Markov measures. Numerical tests on some simple cases with
E
⊂
R
2
show the performance of the proposed methods.</description><subject>Acquisitions & mergers</subject><subject>Algorithms</subject><subject>Analysis</subject><subject>Approximation</subject><subject>Convergence</subject><subject>Markov processes</subject><subject>Mathematical analysis</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Numerical Analysis</subject><subject>Polynomials</subject><issn>0176-4276</issn><issn>1432-0940</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2019</creationdate><recordtype>article</recordtype><recordid>eNp1j71PwzAQRy0EEqEwIzYkZsOd7fhjRBUUpAoYYLYc94JSpU2wk4H_nlRBYmK65b3f6TF2hXCLAOYuA0hdckDLnVLIzRErUEnBwSk4ZgWg0VwJo0_ZWc5bACytNAW7fGvH1PTdQPuhCe31y7ij1MR8zk7q0Ga6-L0L9vH48L584uvX1fPyfs2jRD1waawVNlQanHGuJlFqpbGMUQZSVXCuoo1RAjYmVEJqGakmFUxFTiuDgeSC3cy7feq-RsqD33Zj2k8vvUAHqEvl7EThTMXU5Zyo9n1qdiF9ewR_6Pdzv5_6_aHfm8kRs5Mndv9J6W_5f-kHyT5bsQ</recordid><startdate>20190415</startdate><enddate>20190415</enddate><creator>Piazzon, Federico</creator><general>Springer US</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>20190415</creationdate><title>Pluripotential Numerics</title><author>Piazzon, Federico</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c316t-378828ab609799fe2564615cc3ae4ba99bed7420d7ab2363cefe4a7be96471ae3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2019</creationdate><topic>Acquisitions & mergers</topic><topic>Algorithms</topic><topic>Analysis</topic><topic>Approximation</topic><topic>Convergence</topic><topic>Markov processes</topic><topic>Mathematical analysis</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Numerical Analysis</topic><topic>Polynomials</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Piazzon, Federico</creatorcontrib><collection>CrossRef</collection><jtitle>Constructive approximation</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Piazzon, Federico</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Pluripotential Numerics</atitle><jtitle>Constructive approximation</jtitle><stitle>Constr Approx</stitle><date>2019-04-15</date><risdate>2019</risdate><volume>49</volume><issue>2</issue><spage>227</spage><epage>263</epage><pages>227-263</pages><issn>0176-4276</issn><eissn>1432-0940</eissn><abstract>We study the numerical approximation of the fundamental quantities in
pluripotential theory
, namely the
Siciak Zaharjuta extremal plurisubharmonic function
V
E
∗
of a compact
L
-regular set
E
⊂
C
n
, its
transfinite diameter
δ
(
E
)
,
and the
pluripotential equilibrium measure
μ
E
:
=
dd
c
V
E
∗
n
. The developed methods rely on the computation of a
polynomial mesh
for
E
, for which a suitable orthonormal polynomial basis can be defined. We prove the convergence of the approximation of
δ
(
E
)
and the local uniform convergence of our approximation to
V
E
∗
on
C
n
. Then the convergence of the proposed approximation of
μ
E
follows. Our algorithms are based on the properties of polynomial meshes and Bernstein–Markov measures. Numerical tests on some simple cases with
E
⊂
R
2
show the performance of the proposed methods.</abstract><cop>New York</cop><pub>Springer US</pub><doi>10.1007/s00365-018-9441-7</doi><tpages>37</tpages></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0176-4276 |
| ispartof | Constructive approximation, 2019-04, Vol.49 (2), p.227-263 |
| issn | 0176-4276 1432-0940 |
| language | eng |
| recordid | cdi_proquest_journals_2190165498 |
| source | Springer Nature:Jisc Collections:Springer Nature Read and Publish 2023-2025: Springer Reading List |
| subjects | Acquisitions & mergers Algorithms Analysis Approximation Convergence Markov processes Mathematical analysis Mathematics Mathematics and Statistics Numerical Analysis Polynomials |
| title | Pluripotential Numerics |
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