A Minimum Principle for Potentials with Application to Chebyshev Constants
Abstract For “Riesz-like” kernels K(x,y) = f(|x−y|) on A×A, where A is a compact d-regular set $A\subset \mathbb {R}^{p}$, we prove a minimum principle for potentials $U_{K}^{\mu }=\int K(x,y)\textup {d}\mu (x)$, where μ is a Borel measure supported on A. Setting $P_{K}(\mu )=\inf _{y\in A}U^{\mu }(...
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| Autor*in: |
Reznikov, A. [verfasserIn] |
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| Format: |
Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2017 |
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| Schlagwörter: |
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| Anmerkung: |
© Springer Science+Business Media Dordrecht 2017 |
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| Übergeordnetes Werk: |
Enthalten in: Potential analysis - Springer Netherlands, 1992, 47(2017), 2 vom: 08. Feb., Seite 235-244 |
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| Übergeordnetes Werk: |
volume:47 ; year:2017 ; number:2 ; day:08 ; month:02 ; pages:235-244 |
| Links: |
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| DOI / URN: |
10.1007/s11118-017-9618-x |
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OLC2046684362 |
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| 520 | |a Abstract For “Riesz-like” kernels K(x,y) = f(|x−y|) on A×A, where A is a compact d-regular set $A\subset \mathbb {R}^{p}$, we prove a minimum principle for potentials $U_{K}^{\mu }=\int K(x,y)\textup {d}\mu (x)$, where μ is a Borel measure supported on A. Setting $P_{K}(\mu )=\inf _{y\in A}U^{\mu }(y)$, the K-polarization of μ, the principle is used to show that if {νN} is a sequence of measures on A that converges in the weak-star sense to the measure ν, then PK(νN)→PK(ν) as $N\to \infty $. The continuous Chebyshev (polarization) problem concerns maximizing PK(μ) over all probability measures μ supported on A, while the N-point discrete Chebyshev problem maximizes PK(μ) only over normalized counting measures for N-point multisets on A. We prove for such kernels and sets A, that if {νN} is a sequence of N-point measures solving the discrete problem, then every weak-star limit measure of νN as $N \to \infty $ is a solution to the continuous problem. | ||
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| 650 | 4 | |a Minimum principle | |
| 700 | 1 | |a Saff, E. B. |4 aut | |
| 700 | 1 | |a Vlasiuk, O. V. |4 aut | |
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10.1007/s11118-017-9618-x doi (DE-627)OLC2046684362 (DE-He213)s11118-017-9618-x-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Reznikov, A. verfasserin aut A Minimum Principle for Potentials with Application to Chebyshev Constants 2017 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science+Business Media Dordrecht 2017 Abstract For “Riesz-like” kernels K(x,y) = f(|x−y|) on A×A, where A is a compact d-regular set $A\subset \mathbb {R}^{p}$, we prove a minimum principle for potentials $U_{K}^{\mu }=\int K(x,y)\textup {d}\mu (x)$, where μ is a Borel measure supported on A. Setting $P_{K}(\mu )=\inf _{y\in A}U^{\mu }(y)$, the K-polarization of μ, the principle is used to show that if {νN} is a sequence of measures on A that converges in the weak-star sense to the measure ν, then PK(νN)→PK(ν) as $N\to \infty $. The continuous Chebyshev (polarization) problem concerns maximizing PK(μ) over all probability measures μ supported on A, while the N-point discrete Chebyshev problem maximizes PK(μ) only over normalized counting measures for N-point multisets on A. We prove for such kernels and sets A, that if {νN} is a sequence of N-point measures solving the discrete problem, then every weak-star limit measure of νN as $N \to \infty $ is a solution to the continuous problem. Maximal Riesz polarization Chebyshev constant Hausdorff measure Riesz potential Minimum principle Saff, E. B. aut Vlasiuk, O. V. aut Enthalten in Potential analysis Springer Netherlands, 1992 47(2017), 2 vom: 08. Feb., Seite 235-244 (DE-627)165647787 (DE-600)33485-6 (DE-576)032989911 0926-2601 nnns volume:47 year:2017 number:2 day:08 month:02 pages:235-244 https://doi.org/10.1007/s11118-017-9618-x lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_70 GBV_ILN_2088 AR 47 2017 2 08 02 235-244 |
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10.1007/s11118-017-9618-x doi (DE-627)OLC2046684362 (DE-He213)s11118-017-9618-x-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Reznikov, A. verfasserin aut A Minimum Principle for Potentials with Application to Chebyshev Constants 2017 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science+Business Media Dordrecht 2017 Abstract For “Riesz-like” kernels K(x,y) = f(|x−y|) on A×A, where A is a compact d-regular set $A\subset \mathbb {R}^{p}$, we prove a minimum principle for potentials $U_{K}^{\mu }=\int K(x,y)\textup {d}\mu (x)$, where μ is a Borel measure supported on A. Setting $P_{K}(\mu )=\inf _{y\in A}U^{\mu }(y)$, the K-polarization of μ, the principle is used to show that if {νN} is a sequence of measures on A that converges in the weak-star sense to the measure ν, then PK(νN)→PK(ν) as $N\to \infty $. The continuous Chebyshev (polarization) problem concerns maximizing PK(μ) over all probability measures μ supported on A, while the N-point discrete Chebyshev problem maximizes PK(μ) only over normalized counting measures for N-point multisets on A. We prove for such kernels and sets A, that if {νN} is a sequence of N-point measures solving the discrete problem, then every weak-star limit measure of νN as $N \to \infty $ is a solution to the continuous problem. Maximal Riesz polarization Chebyshev constant Hausdorff measure Riesz potential Minimum principle Saff, E. B. aut Vlasiuk, O. V. aut Enthalten in Potential analysis Springer Netherlands, 1992 47(2017), 2 vom: 08. Feb., Seite 235-244 (DE-627)165647787 (DE-600)33485-6 (DE-576)032989911 0926-2601 nnns volume:47 year:2017 number:2 day:08 month:02 pages:235-244 https://doi.org/10.1007/s11118-017-9618-x lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_70 GBV_ILN_2088 AR 47 2017 2 08 02 235-244 |
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10.1007/s11118-017-9618-x doi (DE-627)OLC2046684362 (DE-He213)s11118-017-9618-x-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Reznikov, A. verfasserin aut A Minimum Principle for Potentials with Application to Chebyshev Constants 2017 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science+Business Media Dordrecht 2017 Abstract For “Riesz-like” kernels K(x,y) = f(|x−y|) on A×A, where A is a compact d-regular set $A\subset \mathbb {R}^{p}$, we prove a minimum principle for potentials $U_{K}^{\mu }=\int K(x,y)\textup {d}\mu (x)$, where μ is a Borel measure supported on A. Setting $P_{K}(\mu )=\inf _{y\in A}U^{\mu }(y)$, the K-polarization of μ, the principle is used to show that if {νN} is a sequence of measures on A that converges in the weak-star sense to the measure ν, then PK(νN)→PK(ν) as $N\to \infty $. The continuous Chebyshev (polarization) problem concerns maximizing PK(μ) over all probability measures μ supported on A, while the N-point discrete Chebyshev problem maximizes PK(μ) only over normalized counting measures for N-point multisets on A. We prove for such kernels and sets A, that if {νN} is a sequence of N-point measures solving the discrete problem, then every weak-star limit measure of νN as $N \to \infty $ is a solution to the continuous problem. Maximal Riesz polarization Chebyshev constant Hausdorff measure Riesz potential Minimum principle Saff, E. B. aut Vlasiuk, O. V. aut Enthalten in Potential analysis Springer Netherlands, 1992 47(2017), 2 vom: 08. Feb., Seite 235-244 (DE-627)165647787 (DE-600)33485-6 (DE-576)032989911 0926-2601 nnns volume:47 year:2017 number:2 day:08 month:02 pages:235-244 https://doi.org/10.1007/s11118-017-9618-x lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_70 GBV_ILN_2088 AR 47 2017 2 08 02 235-244 |
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10.1007/s11118-017-9618-x doi (DE-627)OLC2046684362 (DE-He213)s11118-017-9618-x-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Reznikov, A. verfasserin aut A Minimum Principle for Potentials with Application to Chebyshev Constants 2017 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science+Business Media Dordrecht 2017 Abstract For “Riesz-like” kernels K(x,y) = f(|x−y|) on A×A, where A is a compact d-regular set $A\subset \mathbb {R}^{p}$, we prove a minimum principle for potentials $U_{K}^{\mu }=\int K(x,y)\textup {d}\mu (x)$, where μ is a Borel measure supported on A. Setting $P_{K}(\mu )=\inf _{y\in A}U^{\mu }(y)$, the K-polarization of μ, the principle is used to show that if {νN} is a sequence of measures on A that converges in the weak-star sense to the measure ν, then PK(νN)→PK(ν) as $N\to \infty $. The continuous Chebyshev (polarization) problem concerns maximizing PK(μ) over all probability measures μ supported on A, while the N-point discrete Chebyshev problem maximizes PK(μ) only over normalized counting measures for N-point multisets on A. We prove for such kernels and sets A, that if {νN} is a sequence of N-point measures solving the discrete problem, then every weak-star limit measure of νN as $N \to \infty $ is a solution to the continuous problem. Maximal Riesz polarization Chebyshev constant Hausdorff measure Riesz potential Minimum principle Saff, E. B. aut Vlasiuk, O. V. aut Enthalten in Potential analysis Springer Netherlands, 1992 47(2017), 2 vom: 08. Feb., Seite 235-244 (DE-627)165647787 (DE-600)33485-6 (DE-576)032989911 0926-2601 nnns volume:47 year:2017 number:2 day:08 month:02 pages:235-244 https://doi.org/10.1007/s11118-017-9618-x lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_70 GBV_ILN_2088 AR 47 2017 2 08 02 235-244 |
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10.1007/s11118-017-9618-x doi (DE-627)OLC2046684362 (DE-He213)s11118-017-9618-x-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Reznikov, A. verfasserin aut A Minimum Principle for Potentials with Application to Chebyshev Constants 2017 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science+Business Media Dordrecht 2017 Abstract For “Riesz-like” kernels K(x,y) = f(|x−y|) on A×A, where A is a compact d-regular set $A\subset \mathbb {R}^{p}$, we prove a minimum principle for potentials $U_{K}^{\mu }=\int K(x,y)\textup {d}\mu (x)$, where μ is a Borel measure supported on A. Setting $P_{K}(\mu )=\inf _{y\in A}U^{\mu }(y)$, the K-polarization of μ, the principle is used to show that if {νN} is a sequence of measures on A that converges in the weak-star sense to the measure ν, then PK(νN)→PK(ν) as $N\to \infty $. The continuous Chebyshev (polarization) problem concerns maximizing PK(μ) over all probability measures μ supported on A, while the N-point discrete Chebyshev problem maximizes PK(μ) only over normalized counting measures for N-point multisets on A. We prove for such kernels and sets A, that if {νN} is a sequence of N-point measures solving the discrete problem, then every weak-star limit measure of νN as $N \to \infty $ is a solution to the continuous problem. Maximal Riesz polarization Chebyshev constant Hausdorff measure Riesz potential Minimum principle Saff, E. B. aut Vlasiuk, O. V. aut Enthalten in Potential analysis Springer Netherlands, 1992 47(2017), 2 vom: 08. Feb., Seite 235-244 (DE-627)165647787 (DE-600)33485-6 (DE-576)032989911 0926-2601 nnns volume:47 year:2017 number:2 day:08 month:02 pages:235-244 https://doi.org/10.1007/s11118-017-9618-x lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_70 GBV_ILN_2088 AR 47 2017 2 08 02 235-244 |
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A Minimum Principle for Potentials with Application to Chebyshev Constants |
| abstract |
Abstract For “Riesz-like” kernels K(x,y) = f(|x−y|) on A×A, where A is a compact d-regular set $A\subset \mathbb {R}^{p}$, we prove a minimum principle for potentials $U_{K}^{\mu }=\int K(x,y)\textup {d}\mu (x)$, where μ is a Borel measure supported on A. Setting $P_{K}(\mu )=\inf _{y\in A}U^{\mu }(y)$, the K-polarization of μ, the principle is used to show that if {νN} is a sequence of measures on A that converges in the weak-star sense to the measure ν, then PK(νN)→PK(ν) as $N\to \infty $. The continuous Chebyshev (polarization) problem concerns maximizing PK(μ) over all probability measures μ supported on A, while the N-point discrete Chebyshev problem maximizes PK(μ) only over normalized counting measures for N-point multisets on A. We prove for such kernels and sets A, that if {νN} is a sequence of N-point measures solving the discrete problem, then every weak-star limit measure of νN as $N \to \infty $ is a solution to the continuous problem. © Springer Science+Business Media Dordrecht 2017 |
| abstractGer |
Abstract For “Riesz-like” kernels K(x,y) = f(|x−y|) on A×A, where A is a compact d-regular set $A\subset \mathbb {R}^{p}$, we prove a minimum principle for potentials $U_{K}^{\mu }=\int K(x,y)\textup {d}\mu (x)$, where μ is a Borel measure supported on A. Setting $P_{K}(\mu )=\inf _{y\in A}U^{\mu }(y)$, the K-polarization of μ, the principle is used to show that if {νN} is a sequence of measures on A that converges in the weak-star sense to the measure ν, then PK(νN)→PK(ν) as $N\to \infty $. The continuous Chebyshev (polarization) problem concerns maximizing PK(μ) over all probability measures μ supported on A, while the N-point discrete Chebyshev problem maximizes PK(μ) only over normalized counting measures for N-point multisets on A. We prove for such kernels and sets A, that if {νN} is a sequence of N-point measures solving the discrete problem, then every weak-star limit measure of νN as $N \to \infty $ is a solution to the continuous problem. © Springer Science+Business Media Dordrecht 2017 |
| abstract_unstemmed |
Abstract For “Riesz-like” kernels K(x,y) = f(|x−y|) on A×A, where A is a compact d-regular set $A\subset \mathbb {R}^{p}$, we prove a minimum principle for potentials $U_{K}^{\mu }=\int K(x,y)\textup {d}\mu (x)$, where μ is a Borel measure supported on A. Setting $P_{K}(\mu )=\inf _{y\in A}U^{\mu }(y)$, the K-polarization of μ, the principle is used to show that if {νN} is a sequence of measures on A that converges in the weak-star sense to the measure ν, then PK(νN)→PK(ν) as $N\to \infty $. The continuous Chebyshev (polarization) problem concerns maximizing PK(μ) over all probability measures μ supported on A, while the N-point discrete Chebyshev problem maximizes PK(μ) only over normalized counting measures for N-point multisets on A. We prove for such kernels and sets A, that if {νN} is a sequence of N-point measures solving the discrete problem, then every weak-star limit measure of νN as $N \to \infty $ is a solution to the continuous problem. © Springer Science+Business Media Dordrecht 2017 |
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| title_short |
A Minimum Principle for Potentials with Application to Chebyshev Constants |
| url |
https://doi.org/10.1007/s11118-017-9618-x |
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| author2 |
Saff, E. B. Vlasiuk, O. V. |
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Saff, E. B. Vlasiuk, O. V. |
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| doi_str |
10.1007/s11118-017-9618-x |
| up_date |
2024-07-04T05:39:45.149Z |
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