Tight Gabor Frames Associated with Non-separable Lattices and the Hyperbolic Secant
Abstract We consider Gabor systems generated by a window given by the hyperbolic secant function. We show that such a system forms a Parseval frame for L2(ℝ) when the translations and modulations of the window are associated with certain non-separable lattices in $ ℝ^{2} $ which we explicitly descri...
Ausführliche Beschreibung
Gespeichert in:
| Autor*in: |
Gabardo, Jean-Pierre [verfasserIn] |
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| Format: |
Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2008 |
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| Schlagwörter: |
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| Anmerkung: |
© Springer Science+Business Media B.V. 2008 |
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| Übergeordnetes Werk: |
Enthalten in: Acta applicandae mathematicae - Springer Netherlands, 1983, 107(2008), 1-3 vom: 05. Dez., Seite 49-73 |
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| Übergeordnetes Werk: |
volume:107 ; year:2008 ; number:1-3 ; day:05 ; month:12 ; pages:49-73 |
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| DOI / URN: |
10.1007/s10440-008-9382-4 |
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OLC2070184269 |
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| 520 | |a Abstract We consider Gabor systems generated by a window given by the hyperbolic secant function. We show that such a system forms a Parseval frame for L2(ℝ) when the translations and modulations of the window are associated with certain non-separable lattices in $ ℝ^{2} $ which we explicitly describe. We also study the more general problem of characterizing the positive Borel measures μ on $ ℝ^{2n} $ with the property that the short-time Fourier transform defines an isometric embedding from L2($ ℝ^{n} $) to Lμ2($ ℝ^{2n} $) when the window belongs to the Schwartz class and, in particular, we characterize the extreme points of this set. In the case where the window is the hyperbolic secant function, we consider the situation where the measure is discrete with constant weights and supported on a non-separable lattice yielding a Parseval frame. We provide arithmetic conditions on the parameters defining the lattice characterizing when the associated measure is an extreme point. | ||
| 650 | 4 | |a Irregular Gabor systems | |
| 650 | 4 | |a Translation-bounded measures | |
| 650 | 4 | |a Parseval frames | |
| 650 | 4 | |a Extreme points | |
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10.1007/s10440-008-9382-4 doi (DE-627)OLC2070184269 (DE-He213)s10440-008-9382-4-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Gabardo, Jean-Pierre verfasserin aut Tight Gabor Frames Associated with Non-separable Lattices and the Hyperbolic Secant 2008 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science+Business Media B.V. 2008 Abstract We consider Gabor systems generated by a window given by the hyperbolic secant function. We show that such a system forms a Parseval frame for L2(ℝ) when the translations and modulations of the window are associated with certain non-separable lattices in $ ℝ^{2} $ which we explicitly describe. We also study the more general problem of characterizing the positive Borel measures μ on $ ℝ^{2n} $ with the property that the short-time Fourier transform defines an isometric embedding from L2($ ℝ^{n} $) to Lμ2($ ℝ^{2n} $) when the window belongs to the Schwartz class and, in particular, we characterize the extreme points of this set. In the case where the window is the hyperbolic secant function, we consider the situation where the measure is discrete with constant weights and supported on a non-separable lattice yielding a Parseval frame. We provide arithmetic conditions on the parameters defining the lattice characterizing when the associated measure is an extreme point. Irregular Gabor systems Translation-bounded measures Parseval frames Extreme points Enthalten in Acta applicandae mathematicae Springer Netherlands, 1983 107(2008), 1-3 vom: 05. Dez., Seite 49-73 (DE-627)129132659 (DE-600)46434-X (DE-576)01445274X 0167-8019 nnns volume:107 year:2008 number:1-3 day:05 month:12 pages:49-73 https://doi.org/10.1007/s10440-008-9382-4 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_40 GBV_ILN_70 GBV_ILN_2088 AR 107 2008 1-3 05 12 49-73 |
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10.1007/s10440-008-9382-4 doi (DE-627)OLC2070184269 (DE-He213)s10440-008-9382-4-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Gabardo, Jean-Pierre verfasserin aut Tight Gabor Frames Associated with Non-separable Lattices and the Hyperbolic Secant 2008 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science+Business Media B.V. 2008 Abstract We consider Gabor systems generated by a window given by the hyperbolic secant function. We show that such a system forms a Parseval frame for L2(ℝ) when the translations and modulations of the window are associated with certain non-separable lattices in $ ℝ^{2} $ which we explicitly describe. We also study the more general problem of characterizing the positive Borel measures μ on $ ℝ^{2n} $ with the property that the short-time Fourier transform defines an isometric embedding from L2($ ℝ^{n} $) to Lμ2($ ℝ^{2n} $) when the window belongs to the Schwartz class and, in particular, we characterize the extreme points of this set. In the case where the window is the hyperbolic secant function, we consider the situation where the measure is discrete with constant weights and supported on a non-separable lattice yielding a Parseval frame. We provide arithmetic conditions on the parameters defining the lattice characterizing when the associated measure is an extreme point. Irregular Gabor systems Translation-bounded measures Parseval frames Extreme points Enthalten in Acta applicandae mathematicae Springer Netherlands, 1983 107(2008), 1-3 vom: 05. Dez., Seite 49-73 (DE-627)129132659 (DE-600)46434-X (DE-576)01445274X 0167-8019 nnns volume:107 year:2008 number:1-3 day:05 month:12 pages:49-73 https://doi.org/10.1007/s10440-008-9382-4 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_40 GBV_ILN_70 GBV_ILN_2088 AR 107 2008 1-3 05 12 49-73 |
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10.1007/s10440-008-9382-4 doi (DE-627)OLC2070184269 (DE-He213)s10440-008-9382-4-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Gabardo, Jean-Pierre verfasserin aut Tight Gabor Frames Associated with Non-separable Lattices and the Hyperbolic Secant 2008 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science+Business Media B.V. 2008 Abstract We consider Gabor systems generated by a window given by the hyperbolic secant function. We show that such a system forms a Parseval frame for L2(ℝ) when the translations and modulations of the window are associated with certain non-separable lattices in $ ℝ^{2} $ which we explicitly describe. We also study the more general problem of characterizing the positive Borel measures μ on $ ℝ^{2n} $ with the property that the short-time Fourier transform defines an isometric embedding from L2($ ℝ^{n} $) to Lμ2($ ℝ^{2n} $) when the window belongs to the Schwartz class and, in particular, we characterize the extreme points of this set. In the case where the window is the hyperbolic secant function, we consider the situation where the measure is discrete with constant weights and supported on a non-separable lattice yielding a Parseval frame. We provide arithmetic conditions on the parameters defining the lattice characterizing when the associated measure is an extreme point. Irregular Gabor systems Translation-bounded measures Parseval frames Extreme points Enthalten in Acta applicandae mathematicae Springer Netherlands, 1983 107(2008), 1-3 vom: 05. Dez., Seite 49-73 (DE-627)129132659 (DE-600)46434-X (DE-576)01445274X 0167-8019 nnns volume:107 year:2008 number:1-3 day:05 month:12 pages:49-73 https://doi.org/10.1007/s10440-008-9382-4 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_40 GBV_ILN_70 GBV_ILN_2088 AR 107 2008 1-3 05 12 49-73 |
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10.1007/s10440-008-9382-4 doi (DE-627)OLC2070184269 (DE-He213)s10440-008-9382-4-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Gabardo, Jean-Pierre verfasserin aut Tight Gabor Frames Associated with Non-separable Lattices and the Hyperbolic Secant 2008 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science+Business Media B.V. 2008 Abstract We consider Gabor systems generated by a window given by the hyperbolic secant function. We show that such a system forms a Parseval frame for L2(ℝ) when the translations and modulations of the window are associated with certain non-separable lattices in $ ℝ^{2} $ which we explicitly describe. We also study the more general problem of characterizing the positive Borel measures μ on $ ℝ^{2n} $ with the property that the short-time Fourier transform defines an isometric embedding from L2($ ℝ^{n} $) to Lμ2($ ℝ^{2n} $) when the window belongs to the Schwartz class and, in particular, we characterize the extreme points of this set. In the case where the window is the hyperbolic secant function, we consider the situation where the measure is discrete with constant weights and supported on a non-separable lattice yielding a Parseval frame. We provide arithmetic conditions on the parameters defining the lattice characterizing when the associated measure is an extreme point. Irregular Gabor systems Translation-bounded measures Parseval frames Extreme points Enthalten in Acta applicandae mathematicae Springer Netherlands, 1983 107(2008), 1-3 vom: 05. Dez., Seite 49-73 (DE-627)129132659 (DE-600)46434-X (DE-576)01445274X 0167-8019 nnns volume:107 year:2008 number:1-3 day:05 month:12 pages:49-73 https://doi.org/10.1007/s10440-008-9382-4 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_40 GBV_ILN_70 GBV_ILN_2088 AR 107 2008 1-3 05 12 49-73 |
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10.1007/s10440-008-9382-4 doi (DE-627)OLC2070184269 (DE-He213)s10440-008-9382-4-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Gabardo, Jean-Pierre verfasserin aut Tight Gabor Frames Associated with Non-separable Lattices and the Hyperbolic Secant 2008 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science+Business Media B.V. 2008 Abstract We consider Gabor systems generated by a window given by the hyperbolic secant function. We show that such a system forms a Parseval frame for L2(ℝ) when the translations and modulations of the window are associated with certain non-separable lattices in $ ℝ^{2} $ which we explicitly describe. We also study the more general problem of characterizing the positive Borel measures μ on $ ℝ^{2n} $ with the property that the short-time Fourier transform defines an isometric embedding from L2($ ℝ^{n} $) to Lμ2($ ℝ^{2n} $) when the window belongs to the Schwartz class and, in particular, we characterize the extreme points of this set. In the case where the window is the hyperbolic secant function, we consider the situation where the measure is discrete with constant weights and supported on a non-separable lattice yielding a Parseval frame. We provide arithmetic conditions on the parameters defining the lattice characterizing when the associated measure is an extreme point. Irregular Gabor systems Translation-bounded measures Parseval frames Extreme points Enthalten in Acta applicandae mathematicae Springer Netherlands, 1983 107(2008), 1-3 vom: 05. Dez., Seite 49-73 (DE-627)129132659 (DE-600)46434-X (DE-576)01445274X 0167-8019 nnns volume:107 year:2008 number:1-3 day:05 month:12 pages:49-73 https://doi.org/10.1007/s10440-008-9382-4 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_40 GBV_ILN_70 GBV_ILN_2088 AR 107 2008 1-3 05 12 49-73 |
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Tight Gabor Frames Associated with Non-separable Lattices and the Hyperbolic Secant |
| abstract |
Abstract We consider Gabor systems generated by a window given by the hyperbolic secant function. We show that such a system forms a Parseval frame for L2(ℝ) when the translations and modulations of the window are associated with certain non-separable lattices in $ ℝ^{2} $ which we explicitly describe. We also study the more general problem of characterizing the positive Borel measures μ on $ ℝ^{2n} $ with the property that the short-time Fourier transform defines an isometric embedding from L2($ ℝ^{n} $) to Lμ2($ ℝ^{2n} $) when the window belongs to the Schwartz class and, in particular, we characterize the extreme points of this set. In the case where the window is the hyperbolic secant function, we consider the situation where the measure is discrete with constant weights and supported on a non-separable lattice yielding a Parseval frame. We provide arithmetic conditions on the parameters defining the lattice characterizing when the associated measure is an extreme point. © Springer Science+Business Media B.V. 2008 |
| abstractGer |
Abstract We consider Gabor systems generated by a window given by the hyperbolic secant function. We show that such a system forms a Parseval frame for L2(ℝ) when the translations and modulations of the window are associated with certain non-separable lattices in $ ℝ^{2} $ which we explicitly describe. We also study the more general problem of characterizing the positive Borel measures μ on $ ℝ^{2n} $ with the property that the short-time Fourier transform defines an isometric embedding from L2($ ℝ^{n} $) to Lμ2($ ℝ^{2n} $) when the window belongs to the Schwartz class and, in particular, we characterize the extreme points of this set. In the case where the window is the hyperbolic secant function, we consider the situation where the measure is discrete with constant weights and supported on a non-separable lattice yielding a Parseval frame. We provide arithmetic conditions on the parameters defining the lattice characterizing when the associated measure is an extreme point. © Springer Science+Business Media B.V. 2008 |
| abstract_unstemmed |
Abstract We consider Gabor systems generated by a window given by the hyperbolic secant function. We show that such a system forms a Parseval frame for L2(ℝ) when the translations and modulations of the window are associated with certain non-separable lattices in $ ℝ^{2} $ which we explicitly describe. We also study the more general problem of characterizing the positive Borel measures μ on $ ℝ^{2n} $ with the property that the short-time Fourier transform defines an isometric embedding from L2($ ℝ^{n} $) to Lμ2($ ℝ^{2n} $) when the window belongs to the Schwartz class and, in particular, we characterize the extreme points of this set. In the case where the window is the hyperbolic secant function, we consider the situation where the measure is discrete with constant weights and supported on a non-separable lattice yielding a Parseval frame. We provide arithmetic conditions on the parameters defining the lattice characterizing when the associated measure is an extreme point. © Springer Science+Business Media B.V. 2008 |
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Tight Gabor Frames Associated with Non-separable Lattices and the Hyperbolic Secant |
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