An Operator Based Approach to Irregular Frames of Translates
We consider translates of functions in <inline-formula< <math display="inline"< <semantics< <mrow< <msup< <mi<L</mi< <mn<2</mn< </msup< <mrow< <mo stretchy="false"<(</mo< <msup< <mi mathvariant...
Ausführliche Beschreibung
Autor*in: |
Peter Balazs [verfasserIn] Sigrid Heineken [verfasserIn] |
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Format: |
E-Artikel |
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Sprache: |
Englisch |
Erschienen: |
2019 |
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Schlagwörter: |
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Übergeordnetes Werk: |
In: Mathematics - MDPI AG, 2013, 7(2019), 5, p 449 |
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Übergeordnetes Werk: |
volume:7 ; year:2019 ; number:5, p 449 |
Links: |
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DOI / URN: |
10.3390/math7050449 |
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Katalog-ID: |
DOAJ079622836 |
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10.3390/math7050449 doi (DE-627)DOAJ079622836 (DE-599)DOAJ030d5faa4b4b4cf0a49ad2a0cb104e43 DE-627 ger DE-627 rakwb eng QA1-939 Peter Balazs verfasserin aut An Operator Based Approach to Irregular Frames of Translates 2019 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier We consider translates of functions in <inline-formula< <math display="inline"< <semantics< <mrow< <msup< <mi<L</mi< <mn<2</mn< </msup< <mrow< <mo stretchy="false"<(</mo< <msup< <mi mathvariant="double-struck"<R</mi< <mi<d</mi< </msup< <mo stretchy="false"<)</mo< </mrow< </mrow< </semantics< </math< </inline-formula< along an irregular set of points, that is, <inline-formula< <math display="inline"< <semantics< <msub< <mrow< <mo stretchy="false"<{</mo< <mi<ϕ</mi< <mrow< <mo stretchy="false"<(</mo< <mo<·</mo< <mo<−</mo< <msub< <mi<λ</mi< <mi<k</mi< </msub< <mo stretchy="false"<)</mo< </mrow< <mo stretchy="false"<}</mo< </mrow< <mrow< <mi<k</mi< <mo<∈</mo< <mi mathvariant="double-struck"<Z</mi< </mrow< </msub< </semantics< </math< </inline-formula<—where <inline-formula< <math display="inline"< <semantics< <mi<ϕ</mi< </semantics< </math< </inline-formula< is a bandlimited function. Introducing a notion of pseudo-Gramian function for the irregular case, we obtain conditions for a family of irregular translates to be a Bessel, frame or Riesz sequence. We show the connection of the frame-related operators of the translates to the operators of exponentials. This is used, in particular, to find for the first time in the irregular case a representation of the canonical dual as well as of the equivalent Parseval frame—in terms of its Fourier transform. frames Riesz bases irregular translates canonical duals frame-related operators Mathematics Sigrid Heineken verfasserin aut In Mathematics MDPI AG, 2013 7(2019), 5, p 449 (DE-627)737287764 (DE-600)2704244-3 22277390 nnns volume:7 year:2019 number:5, p 449 https://doi.org/10.3390/math7050449 kostenfrei https://doaj.org/article/030d5faa4b4b4cf0a49ad2a0cb104e43 kostenfrei https://www.mdpi.com/2227-7390/7/5/449 kostenfrei https://doaj.org/toc/2227-7390 Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_2005 GBV_ILN_2009 GBV_ILN_2014 GBV_ILN_2055 GBV_ILN_2111 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 7 2019 5, p 449 |
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10.3390/math7050449 doi (DE-627)DOAJ079622836 (DE-599)DOAJ030d5faa4b4b4cf0a49ad2a0cb104e43 DE-627 ger DE-627 rakwb eng QA1-939 Peter Balazs verfasserin aut An Operator Based Approach to Irregular Frames of Translates 2019 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier We consider translates of functions in <inline-formula< <math display="inline"< <semantics< <mrow< <msup< <mi<L</mi< <mn<2</mn< </msup< <mrow< <mo stretchy="false"<(</mo< <msup< <mi mathvariant="double-struck"<R</mi< <mi<d</mi< </msup< <mo stretchy="false"<)</mo< </mrow< </mrow< </semantics< </math< </inline-formula< along an irregular set of points, that is, <inline-formula< <math display="inline"< <semantics< <msub< <mrow< <mo stretchy="false"<{</mo< <mi<ϕ</mi< <mrow< <mo stretchy="false"<(</mo< <mo<·</mo< <mo<−</mo< <msub< <mi<λ</mi< <mi<k</mi< </msub< <mo stretchy="false"<)</mo< </mrow< <mo stretchy="false"<}</mo< </mrow< <mrow< <mi<k</mi< <mo<∈</mo< <mi mathvariant="double-struck"<Z</mi< </mrow< </msub< </semantics< </math< </inline-formula<—where <inline-formula< <math display="inline"< <semantics< <mi<ϕ</mi< </semantics< </math< </inline-formula< is a bandlimited function. Introducing a notion of pseudo-Gramian function for the irregular case, we obtain conditions for a family of irregular translates to be a Bessel, frame or Riesz sequence. We show the connection of the frame-related operators of the translates to the operators of exponentials. This is used, in particular, to find for the first time in the irregular case a representation of the canonical dual as well as of the equivalent Parseval frame—in terms of its Fourier transform. frames Riesz bases irregular translates canonical duals frame-related operators Mathematics Sigrid Heineken verfasserin aut In Mathematics MDPI AG, 2013 7(2019), 5, p 449 (DE-627)737287764 (DE-600)2704244-3 22277390 nnns volume:7 year:2019 number:5, p 449 https://doi.org/10.3390/math7050449 kostenfrei https://doaj.org/article/030d5faa4b4b4cf0a49ad2a0cb104e43 kostenfrei https://www.mdpi.com/2227-7390/7/5/449 kostenfrei https://doaj.org/toc/2227-7390 Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_2005 GBV_ILN_2009 GBV_ILN_2014 GBV_ILN_2055 GBV_ILN_2111 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 7 2019 5, p 449 |
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10.3390/math7050449 doi (DE-627)DOAJ079622836 (DE-599)DOAJ030d5faa4b4b4cf0a49ad2a0cb104e43 DE-627 ger DE-627 rakwb eng QA1-939 Peter Balazs verfasserin aut An Operator Based Approach to Irregular Frames of Translates 2019 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier We consider translates of functions in <inline-formula< <math display="inline"< <semantics< <mrow< <msup< <mi<L</mi< <mn<2</mn< </msup< <mrow< <mo stretchy="false"<(</mo< <msup< <mi mathvariant="double-struck"<R</mi< <mi<d</mi< </msup< <mo stretchy="false"<)</mo< </mrow< </mrow< </semantics< </math< </inline-formula< along an irregular set of points, that is, <inline-formula< <math display="inline"< <semantics< <msub< <mrow< <mo stretchy="false"<{</mo< <mi<ϕ</mi< <mrow< <mo stretchy="false"<(</mo< <mo<·</mo< <mo<−</mo< <msub< <mi<λ</mi< <mi<k</mi< </msub< <mo stretchy="false"<)</mo< </mrow< <mo stretchy="false"<}</mo< </mrow< <mrow< <mi<k</mi< <mo<∈</mo< <mi mathvariant="double-struck"<Z</mi< </mrow< </msub< </semantics< </math< </inline-formula<—where <inline-formula< <math display="inline"< <semantics< <mi<ϕ</mi< </semantics< </math< </inline-formula< is a bandlimited function. Introducing a notion of pseudo-Gramian function for the irregular case, we obtain conditions for a family of irregular translates to be a Bessel, frame or Riesz sequence. We show the connection of the frame-related operators of the translates to the operators of exponentials. This is used, in particular, to find for the first time in the irregular case a representation of the canonical dual as well as of the equivalent Parseval frame—in terms of its Fourier transform. frames Riesz bases irregular translates canonical duals frame-related operators Mathematics Sigrid Heineken verfasserin aut In Mathematics MDPI AG, 2013 7(2019), 5, p 449 (DE-627)737287764 (DE-600)2704244-3 22277390 nnns volume:7 year:2019 number:5, p 449 https://doi.org/10.3390/math7050449 kostenfrei https://doaj.org/article/030d5faa4b4b4cf0a49ad2a0cb104e43 kostenfrei https://www.mdpi.com/2227-7390/7/5/449 kostenfrei https://doaj.org/toc/2227-7390 Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_2005 GBV_ILN_2009 GBV_ILN_2014 GBV_ILN_2055 GBV_ILN_2111 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 7 2019 5, p 449 |
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10.3390/math7050449 doi (DE-627)DOAJ079622836 (DE-599)DOAJ030d5faa4b4b4cf0a49ad2a0cb104e43 DE-627 ger DE-627 rakwb eng QA1-939 Peter Balazs verfasserin aut An Operator Based Approach to Irregular Frames of Translates 2019 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier We consider translates of functions in <inline-formula< <math display="inline"< <semantics< <mrow< <msup< <mi<L</mi< <mn<2</mn< </msup< <mrow< <mo stretchy="false"<(</mo< <msup< <mi mathvariant="double-struck"<R</mi< <mi<d</mi< </msup< <mo stretchy="false"<)</mo< </mrow< </mrow< </semantics< </math< </inline-formula< along an irregular set of points, that is, <inline-formula< <math display="inline"< <semantics< <msub< <mrow< <mo stretchy="false"<{</mo< <mi<ϕ</mi< <mrow< <mo stretchy="false"<(</mo< <mo<·</mo< <mo<−</mo< <msub< <mi<λ</mi< <mi<k</mi< </msub< <mo stretchy="false"<)</mo< </mrow< <mo stretchy="false"<}</mo< </mrow< <mrow< <mi<k</mi< <mo<∈</mo< <mi mathvariant="double-struck"<Z</mi< </mrow< </msub< </semantics< </math< </inline-formula<—where <inline-formula< <math display="inline"< <semantics< <mi<ϕ</mi< </semantics< </math< </inline-formula< is a bandlimited function. Introducing a notion of pseudo-Gramian function for the irregular case, we obtain conditions for a family of irregular translates to be a Bessel, frame or Riesz sequence. We show the connection of the frame-related operators of the translates to the operators of exponentials. This is used, in particular, to find for the first time in the irregular case a representation of the canonical dual as well as of the equivalent Parseval frame—in terms of its Fourier transform. frames Riesz bases irregular translates canonical duals frame-related operators Mathematics Sigrid Heineken verfasserin aut In Mathematics MDPI AG, 2013 7(2019), 5, p 449 (DE-627)737287764 (DE-600)2704244-3 22277390 nnns volume:7 year:2019 number:5, p 449 https://doi.org/10.3390/math7050449 kostenfrei https://doaj.org/article/030d5faa4b4b4cf0a49ad2a0cb104e43 kostenfrei https://www.mdpi.com/2227-7390/7/5/449 kostenfrei https://doaj.org/toc/2227-7390 Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_2005 GBV_ILN_2009 GBV_ILN_2014 GBV_ILN_2055 GBV_ILN_2111 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 7 2019 5, p 449 |
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English |
source |
In Mathematics 7(2019), 5, p 449 volume:7 year:2019 number:5, p 449 |
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In Mathematics 7(2019), 5, p 449 volume:7 year:2019 number:5, p 449 |
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Article |
institution |
findex.gbv.de |
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An Operator Based Approach to Irregular Frames of Translates |
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We consider translates of functions in <inline-formula< <math display="inline"< <semantics< <mrow< <msup< <mi<L</mi< <mn<2</mn< </msup< <mrow< <mo stretchy="false"<(</mo< <msup< <mi mathvariant="double-struck"<R</mi< <mi<d</mi< </msup< <mo stretchy="false"<)</mo< </mrow< </mrow< </semantics< </math< </inline-formula< along an irregular set of points, that is, <inline-formula< <math display="inline"< <semantics< <msub< <mrow< <mo stretchy="false"<{</mo< <mi<ϕ</mi< <mrow< <mo stretchy="false"<(</mo< <mo<·</mo< <mo<−</mo< <msub< <mi<λ</mi< <mi<k</mi< </msub< <mo stretchy="false"<)</mo< </mrow< <mo stretchy="false"<}</mo< </mrow< <mrow< <mi<k</mi< <mo<∈</mo< <mi mathvariant="double-struck"<Z</mi< </mrow< </msub< </semantics< </math< </inline-formula<—where <inline-formula< <math display="inline"< <semantics< <mi<ϕ</mi< </semantics< </math< </inline-formula< is a bandlimited function. Introducing a notion of pseudo-Gramian function for the irregular case, we obtain conditions for a family of irregular translates to be a Bessel, frame or Riesz sequence. We show the connection of the frame-related operators of the translates to the operators of exponentials. This is used, in particular, to find for the first time in the irregular case a representation of the canonical dual as well as of the equivalent Parseval frame—in terms of its Fourier transform. |
abstractGer |
We consider translates of functions in <inline-formula< <math display="inline"< <semantics< <mrow< <msup< <mi<L</mi< <mn<2</mn< </msup< <mrow< <mo stretchy="false"<(</mo< <msup< <mi mathvariant="double-struck"<R</mi< <mi<d</mi< </msup< <mo stretchy="false"<)</mo< </mrow< </mrow< </semantics< </math< </inline-formula< along an irregular set of points, that is, <inline-formula< <math display="inline"< <semantics< <msub< <mrow< <mo stretchy="false"<{</mo< <mi<ϕ</mi< <mrow< <mo stretchy="false"<(</mo< <mo<·</mo< <mo<−</mo< <msub< <mi<λ</mi< <mi<k</mi< </msub< <mo stretchy="false"<)</mo< </mrow< <mo stretchy="false"<}</mo< </mrow< <mrow< <mi<k</mi< <mo<∈</mo< <mi mathvariant="double-struck"<Z</mi< </mrow< </msub< </semantics< </math< </inline-formula<—where <inline-formula< <math display="inline"< <semantics< <mi<ϕ</mi< </semantics< </math< </inline-formula< is a bandlimited function. Introducing a notion of pseudo-Gramian function for the irregular case, we obtain conditions for a family of irregular translates to be a Bessel, frame or Riesz sequence. We show the connection of the frame-related operators of the translates to the operators of exponentials. This is used, in particular, to find for the first time in the irregular case a representation of the canonical dual as well as of the equivalent Parseval frame—in terms of its Fourier transform. |
abstract_unstemmed |
We consider translates of functions in <inline-formula< <math display="inline"< <semantics< <mrow< <msup< <mi<L</mi< <mn<2</mn< </msup< <mrow< <mo stretchy="false"<(</mo< <msup< <mi mathvariant="double-struck"<R</mi< <mi<d</mi< </msup< <mo stretchy="false"<)</mo< </mrow< </mrow< </semantics< </math< </inline-formula< along an irregular set of points, that is, <inline-formula< <math display="inline"< <semantics< <msub< <mrow< <mo stretchy="false"<{</mo< <mi<ϕ</mi< <mrow< <mo stretchy="false"<(</mo< <mo<·</mo< <mo<−</mo< <msub< <mi<λ</mi< <mi<k</mi< </msub< <mo stretchy="false"<)</mo< </mrow< <mo stretchy="false"<}</mo< </mrow< <mrow< <mi<k</mi< <mo<∈</mo< <mi mathvariant="double-struck"<Z</mi< </mrow< </msub< </semantics< </math< </inline-formula<—where <inline-formula< <math display="inline"< <semantics< <mi<ϕ</mi< </semantics< </math< </inline-formula< is a bandlimited function. Introducing a notion of pseudo-Gramian function for the irregular case, we obtain conditions for a family of irregular translates to be a Bessel, frame or Riesz sequence. We show the connection of the frame-related operators of the translates to the operators of exponentials. This is used, in particular, to find for the first time in the irregular case a representation of the canonical dual as well as of the equivalent Parseval frame—in terms of its Fourier transform. |
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