A note on exponential Riesz bases
Abstract We prove that if %$I_\ell = [a_\ell ,b_\ell )%$, %$\ell =1,\ldots ,L%$, are disjoint intervals in [0, 1) with the property that the numbers %$1, a_1, \ldots , a_L, b_1, \ldots , b_L%$ are linearly independent over %${\mathbb {Q}}%$, then there exist pairwise disjoint sets %$\Lambda _\ell \s...
Ausführliche Beschreibung
Gespeichert in:
| Autor*in: |
Caragea, Andrei [verfasserIn] |
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| Format: |
E-Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2022 |
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| Schlagwörter: |
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| Anmerkung: |
© The Author(s) 2022 |
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| Übergeordnetes Werk: |
Enthalten in: Sampling theory, signal processing, and data analysis - [Cham] : Birkhäuser, 2021, 20(2022), 2 vom: 28. Juli |
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| Übergeordnetes Werk: |
volume:20 ; year:2022 ; number:2 ; day:28 ; month:07 |
| Links: |
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| DOI / URN: |
10.1007/s43670-022-00031-9 |
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| Katalog-ID: |
SPR047710616 |
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| 100 | 1 | |a Caragea, Andrei |e verfasserin |4 aut | |
| 245 | 1 | 2 | |a A note on exponential Riesz bases |
| 264 | 1 | |c 2022 | |
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| 520 | |a Abstract We prove that if %$I_\ell = [a_\ell ,b_\ell )%$, %$\ell =1,\ldots ,L%$, are disjoint intervals in [0, 1) with the property that the numbers %$1, a_1, \ldots , a_L, b_1, \ldots , b_L%$ are linearly independent over %${\mathbb {Q}}%$, then there exist pairwise disjoint sets %$\Lambda _\ell \subset {\mathbb {Z}}%$, %$\ell =1, \ldots , L%$, such that for every %$J \subset \{ 1, \ldots , L \}%$, the system %$\{e^{2\pi i \lambda x} : \lambda \in \cup _{\ell \in J} \, \Lambda _\ell \}%$ is a Riesz basis for %$L^2 ( \cup _{\ell \in J} \, I_\ell )%$. Also, we show that for any disjoint intervals %$I_\ell %$, %$\ell =1, \ldots , L%$, contained in [1, N) with %$N \in {\mathbb {N}}%$, the orthonormal basis %$\{e^{2\pi i n x} : n \in {\mathbb {Z}}\}%$ of %$L^2[0,1)%$ can be complemented by a Riesz basis %$\{e^{2\pi i \lambda x}: \lambda \in \Lambda \}%$ for %$L^2(\cup _{\ell =1}^L \, I_{\ell })%$ with some set %$\Lambda \subset (\frac{1}{N} {\mathbb {Z}}) \backslash {\mathbb {Z}}%$, in the sense that their union %$\{e^{2\pi i \lambda x} : \lambda \in {\mathbb {Z}}\cup \Lambda \}%$ is a Riesz basis for %$L^2 ( [0,1) \cup I_1 \cup \cdots \cup I_L )%$. | ||
| 650 | 4 | |a Exponential bases |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a Riesz bases |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a Hierarchical structure |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a Finite union of intervals |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a Kronecker–Weyl equidistribution along the primes |7 (dpeaa)DE-He213 | |
| 700 | 1 | |a Lee, Dae Gwan |0 (orcid)0000-0002-0991-738X |4 aut | |
| 773 | 0 | 8 | |i Enthalten in |t Sampling theory, signal processing, and data analysis |d [Cham] : Birkhäuser, 2021 |g 20(2022), 2 vom: 28. Juli |w (DE-627)1735681601 |w (DE-600)3041928-1 |x 2730-5724 |7 nnns |
| 773 | 1 | 8 | |g volume:20 |g year:2022 |g number:2 |g day:28 |g month:07 |
| 856 | 4 | 0 | |u https://dx.doi.org/10.1007/s43670-022-00031-9 |z kostenfrei |3 Volltext |
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10.1007/s43670-022-00031-9 doi (DE-627)SPR047710616 (SPR)s43670-022-00031-9-e DE-627 ger DE-627 rakwb eng Caragea, Andrei verfasserin aut A note on exponential Riesz bases 2022 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s) 2022 Abstract We prove that if %$I_\ell = [a_\ell ,b_\ell )%$, %$\ell =1,\ldots ,L%$, are disjoint intervals in [0, 1) with the property that the numbers %$1, a_1, \ldots , a_L, b_1, \ldots , b_L%$ are linearly independent over %${\mathbb {Q}}%$, then there exist pairwise disjoint sets %$\Lambda _\ell \subset {\mathbb {Z}}%$, %$\ell =1, \ldots , L%$, such that for every %$J \subset \{ 1, \ldots , L \}%$, the system %$\{e^{2\pi i \lambda x} : \lambda \in \cup _{\ell \in J} \, \Lambda _\ell \}%$ is a Riesz basis for %$L^2 ( \cup _{\ell \in J} \, I_\ell )%$. Also, we show that for any disjoint intervals %$I_\ell %$, %$\ell =1, \ldots , L%$, contained in [1, N) with %$N \in {\mathbb {N}}%$, the orthonormal basis %$\{e^{2\pi i n x} : n \in {\mathbb {Z}}\}%$ of %$L^2[0,1)%$ can be complemented by a Riesz basis %$\{e^{2\pi i \lambda x}: \lambda \in \Lambda \}%$ for %$L^2(\cup _{\ell =1}^L \, I_{\ell })%$ with some set %$\Lambda \subset (\frac{1}{N} {\mathbb {Z}}) \backslash {\mathbb {Z}}%$, in the sense that their union %$\{e^{2\pi i \lambda x} : \lambda \in {\mathbb {Z}}\cup \Lambda \}%$ is a Riesz basis for %$L^2 ( [0,1) \cup I_1 \cup \cdots \cup I_L )%$. Exponential bases (dpeaa)DE-He213 Riesz bases (dpeaa)DE-He213 Hierarchical structure (dpeaa)DE-He213 Finite union of intervals (dpeaa)DE-He213 Kronecker–Weyl equidistribution along the primes (dpeaa)DE-He213 Lee, Dae Gwan (orcid)0000-0002-0991-738X aut Enthalten in Sampling theory, signal processing, and data analysis [Cham] : Birkhäuser, 2021 20(2022), 2 vom: 28. Juli (DE-627)1735681601 (DE-600)3041928-1 2730-5724 nnns volume:20 year:2022 number:2 day:28 month:07 https://dx.doi.org/10.1007/s43670-022-00031-9 kostenfrei Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 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_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_101 GBV_ILN_105 GBV_ILN_110 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 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_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 20 2022 2 28 07 |
| spelling |
10.1007/s43670-022-00031-9 doi (DE-627)SPR047710616 (SPR)s43670-022-00031-9-e DE-627 ger DE-627 rakwb eng Caragea, Andrei verfasserin aut A note on exponential Riesz bases 2022 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s) 2022 Abstract We prove that if %$I_\ell = [a_\ell ,b_\ell )%$, %$\ell =1,\ldots ,L%$, are disjoint intervals in [0, 1) with the property that the numbers %$1, a_1, \ldots , a_L, b_1, \ldots , b_L%$ are linearly independent over %${\mathbb {Q}}%$, then there exist pairwise disjoint sets %$\Lambda _\ell \subset {\mathbb {Z}}%$, %$\ell =1, \ldots , L%$, such that for every %$J \subset \{ 1, \ldots , L \}%$, the system %$\{e^{2\pi i \lambda x} : \lambda \in \cup _{\ell \in J} \, \Lambda _\ell \}%$ is a Riesz basis for %$L^2 ( \cup _{\ell \in J} \, I_\ell )%$. Also, we show that for any disjoint intervals %$I_\ell %$, %$\ell =1, \ldots , L%$, contained in [1, N) with %$N \in {\mathbb {N}}%$, the orthonormal basis %$\{e^{2\pi i n x} : n \in {\mathbb {Z}}\}%$ of %$L^2[0,1)%$ can be complemented by a Riesz basis %$\{e^{2\pi i \lambda x}: \lambda \in \Lambda \}%$ for %$L^2(\cup _{\ell =1}^L \, I_{\ell })%$ with some set %$\Lambda \subset (\frac{1}{N} {\mathbb {Z}}) \backslash {\mathbb {Z}}%$, in the sense that their union %$\{e^{2\pi i \lambda x} : \lambda \in {\mathbb {Z}}\cup \Lambda \}%$ is a Riesz basis for %$L^2 ( [0,1) \cup I_1 \cup \cdots \cup I_L )%$. Exponential bases (dpeaa)DE-He213 Riesz bases (dpeaa)DE-He213 Hierarchical structure (dpeaa)DE-He213 Finite union of intervals (dpeaa)DE-He213 Kronecker–Weyl equidistribution along the primes (dpeaa)DE-He213 Lee, Dae Gwan (orcid)0000-0002-0991-738X aut Enthalten in Sampling theory, signal processing, and data analysis [Cham] : Birkhäuser, 2021 20(2022), 2 vom: 28. Juli (DE-627)1735681601 (DE-600)3041928-1 2730-5724 nnns volume:20 year:2022 number:2 day:28 month:07 https://dx.doi.org/10.1007/s43670-022-00031-9 kostenfrei Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 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_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_101 GBV_ILN_105 GBV_ILN_110 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 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_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 20 2022 2 28 07 |
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10.1007/s43670-022-00031-9 doi (DE-627)SPR047710616 (SPR)s43670-022-00031-9-e DE-627 ger DE-627 rakwb eng Caragea, Andrei verfasserin aut A note on exponential Riesz bases 2022 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s) 2022 Abstract We prove that if %$I_\ell = [a_\ell ,b_\ell )%$, %$\ell =1,\ldots ,L%$, are disjoint intervals in [0, 1) with the property that the numbers %$1, a_1, \ldots , a_L, b_1, \ldots , b_L%$ are linearly independent over %${\mathbb {Q}}%$, then there exist pairwise disjoint sets %$\Lambda _\ell \subset {\mathbb {Z}}%$, %$\ell =1, \ldots , L%$, such that for every %$J \subset \{ 1, \ldots , L \}%$, the system %$\{e^{2\pi i \lambda x} : \lambda \in \cup _{\ell \in J} \, \Lambda _\ell \}%$ is a Riesz basis for %$L^2 ( \cup _{\ell \in J} \, I_\ell )%$. Also, we show that for any disjoint intervals %$I_\ell %$, %$\ell =1, \ldots , L%$, contained in [1, N) with %$N \in {\mathbb {N}}%$, the orthonormal basis %$\{e^{2\pi i n x} : n \in {\mathbb {Z}}\}%$ of %$L^2[0,1)%$ can be complemented by a Riesz basis %$\{e^{2\pi i \lambda x}: \lambda \in \Lambda \}%$ for %$L^2(\cup _{\ell =1}^L \, I_{\ell })%$ with some set %$\Lambda \subset (\frac{1}{N} {\mathbb {Z}}) \backslash {\mathbb {Z}}%$, in the sense that their union %$\{e^{2\pi i \lambda x} : \lambda \in {\mathbb {Z}}\cup \Lambda \}%$ is a Riesz basis for %$L^2 ( [0,1) \cup I_1 \cup \cdots \cup I_L )%$. Exponential bases (dpeaa)DE-He213 Riesz bases (dpeaa)DE-He213 Hierarchical structure (dpeaa)DE-He213 Finite union of intervals (dpeaa)DE-He213 Kronecker–Weyl equidistribution along the primes (dpeaa)DE-He213 Lee, Dae Gwan (orcid)0000-0002-0991-738X aut Enthalten in Sampling theory, signal processing, and data analysis [Cham] : Birkhäuser, 2021 20(2022), 2 vom: 28. Juli (DE-627)1735681601 (DE-600)3041928-1 2730-5724 nnns volume:20 year:2022 number:2 day:28 month:07 https://dx.doi.org/10.1007/s43670-022-00031-9 kostenfrei Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 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_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_101 GBV_ILN_105 GBV_ILN_110 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 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_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 20 2022 2 28 07 |
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10.1007/s43670-022-00031-9 doi (DE-627)SPR047710616 (SPR)s43670-022-00031-9-e DE-627 ger DE-627 rakwb eng Caragea, Andrei verfasserin aut A note on exponential Riesz bases 2022 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s) 2022 Abstract We prove that if %$I_\ell = [a_\ell ,b_\ell )%$, %$\ell =1,\ldots ,L%$, are disjoint intervals in [0, 1) with the property that the numbers %$1, a_1, \ldots , a_L, b_1, \ldots , b_L%$ are linearly independent over %${\mathbb {Q}}%$, then there exist pairwise disjoint sets %$\Lambda _\ell \subset {\mathbb {Z}}%$, %$\ell =1, \ldots , L%$, such that for every %$J \subset \{ 1, \ldots , L \}%$, the system %$\{e^{2\pi i \lambda x} : \lambda \in \cup _{\ell \in J} \, \Lambda _\ell \}%$ is a Riesz basis for %$L^2 ( \cup _{\ell \in J} \, I_\ell )%$. Also, we show that for any disjoint intervals %$I_\ell %$, %$\ell =1, \ldots , L%$, contained in [1, N) with %$N \in {\mathbb {N}}%$, the orthonormal basis %$\{e^{2\pi i n x} : n \in {\mathbb {Z}}\}%$ of %$L^2[0,1)%$ can be complemented by a Riesz basis %$\{e^{2\pi i \lambda x}: \lambda \in \Lambda \}%$ for %$L^2(\cup _{\ell =1}^L \, I_{\ell })%$ with some set %$\Lambda \subset (\frac{1}{N} {\mathbb {Z}}) \backslash {\mathbb {Z}}%$, in the sense that their union %$\{e^{2\pi i \lambda x} : \lambda \in {\mathbb {Z}}\cup \Lambda \}%$ is a Riesz basis for %$L^2 ( [0,1) \cup I_1 \cup \cdots \cup I_L )%$. Exponential bases (dpeaa)DE-He213 Riesz bases (dpeaa)DE-He213 Hierarchical structure (dpeaa)DE-He213 Finite union of intervals (dpeaa)DE-He213 Kronecker–Weyl equidistribution along the primes (dpeaa)DE-He213 Lee, Dae Gwan (orcid)0000-0002-0991-738X aut Enthalten in Sampling theory, signal processing, and data analysis [Cham] : Birkhäuser, 2021 20(2022), 2 vom: 28. Juli (DE-627)1735681601 (DE-600)3041928-1 2730-5724 nnns volume:20 year:2022 number:2 day:28 month:07 https://dx.doi.org/10.1007/s43670-022-00031-9 kostenfrei Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 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_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_101 GBV_ILN_105 GBV_ILN_110 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 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_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 20 2022 2 28 07 |
| allfieldsSound |
10.1007/s43670-022-00031-9 doi (DE-627)SPR047710616 (SPR)s43670-022-00031-9-e DE-627 ger DE-627 rakwb eng Caragea, Andrei verfasserin aut A note on exponential Riesz bases 2022 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s) 2022 Abstract We prove that if %$I_\ell = [a_\ell ,b_\ell )%$, %$\ell =1,\ldots ,L%$, are disjoint intervals in [0, 1) with the property that the numbers %$1, a_1, \ldots , a_L, b_1, \ldots , b_L%$ are linearly independent over %${\mathbb {Q}}%$, then there exist pairwise disjoint sets %$\Lambda _\ell \subset {\mathbb {Z}}%$, %$\ell =1, \ldots , L%$, such that for every %$J \subset \{ 1, \ldots , L \}%$, the system %$\{e^{2\pi i \lambda x} : \lambda \in \cup _{\ell \in J} \, \Lambda _\ell \}%$ is a Riesz basis for %$L^2 ( \cup _{\ell \in J} \, I_\ell )%$. Also, we show that for any disjoint intervals %$I_\ell %$, %$\ell =1, \ldots , L%$, contained in [1, N) with %$N \in {\mathbb {N}}%$, the orthonormal basis %$\{e^{2\pi i n x} : n \in {\mathbb {Z}}\}%$ of %$L^2[0,1)%$ can be complemented by a Riesz basis %$\{e^{2\pi i \lambda x}: \lambda \in \Lambda \}%$ for %$L^2(\cup _{\ell =1}^L \, I_{\ell })%$ with some set %$\Lambda \subset (\frac{1}{N} {\mathbb {Z}}) \backslash {\mathbb {Z}}%$, in the sense that their union %$\{e^{2\pi i \lambda x} : \lambda \in {\mathbb {Z}}\cup \Lambda \}%$ is a Riesz basis for %$L^2 ( [0,1) \cup I_1 \cup \cdots \cup I_L )%$. Exponential bases (dpeaa)DE-He213 Riesz bases (dpeaa)DE-He213 Hierarchical structure (dpeaa)DE-He213 Finite union of intervals (dpeaa)DE-He213 Kronecker–Weyl equidistribution along the primes (dpeaa)DE-He213 Lee, Dae Gwan (orcid)0000-0002-0991-738X aut Enthalten in Sampling theory, signal processing, and data analysis [Cham] : Birkhäuser, 2021 20(2022), 2 vom: 28. Juli (DE-627)1735681601 (DE-600)3041928-1 2730-5724 nnns volume:20 year:2022 number:2 day:28 month:07 https://dx.doi.org/10.1007/s43670-022-00031-9 kostenfrei Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 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_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_101 GBV_ILN_105 GBV_ILN_110 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 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_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 20 2022 2 28 07 |
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English |
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Enthalten in Sampling theory, signal processing, and data analysis 20(2022), 2 vom: 28. Juli volume:20 year:2022 number:2 day:28 month:07 |
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Enthalten in Sampling theory, signal processing, and data analysis 20(2022), 2 vom: 28. Juli volume:20 year:2022 number:2 day:28 month:07 |
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Exponential bases Riesz bases Hierarchical structure Finite union of intervals Kronecker–Weyl equidistribution along the primes |
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Sampling theory, signal processing, and data analysis |
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Caragea, Andrei @@aut@@ Lee, Dae Gwan @@aut@@ |
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2022-07-28T00:00:00Z |
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Also, we show that for any disjoint intervals %$I_\ell %$, %$\ell =1, \ldots , L%$, contained in [1, N) with %$N \in {\mathbb {N}}%$, the orthonormal basis %$\{e^{2\pi i n x} : n \in {\mathbb {Z}}\}%$ of %$L^2[0,1)%$ can be complemented by a Riesz basis %$\{e^{2\pi i \lambda x}: \lambda \in \Lambda \}%$ for %$L^2(\cup _{\ell =1}^L \, I_{\ell })%$ with some set %$\Lambda \subset (\frac{1}{N} {\mathbb {Z}}) \backslash {\mathbb {Z}}%$, in the sense that their union %$\{e^{2\pi i \lambda x} : \lambda \in {\mathbb {Z}}\cup \Lambda \}%$ is a Riesz basis for %$L^2 ( [0,1) \cup I_1 \cup \cdots \cup I_L )%$.</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Exponential bases</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Riesz bases</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Hierarchical structure</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Finite union of intervals</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Kronecker–Weyl equidistribution along the primes</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="700" ind1="1" ind2=" "><subfield code="a">Lee, Dae Gwan</subfield><subfield code="0">(orcid)0000-0002-0991-738X</subfield><subfield code="4">aut</subfield></datafield><datafield tag="773" ind1="0" ind2="8"><subfield code="i">Enthalten in</subfield><subfield code="t">Sampling theory, signal processing, and data analysis</subfield><subfield code="d">[Cham] : Birkhäuser, 2021</subfield><subfield code="g">20(2022), 2 vom: 28. 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Caragea, Andrei |
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Caragea, Andrei misc Exponential bases misc Riesz bases misc Hierarchical structure misc Finite union of intervals misc Kronecker–Weyl equidistribution along the primes A note on exponential Riesz bases |
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A note on exponential Riesz bases Exponential bases (dpeaa)DE-He213 Riesz bases (dpeaa)DE-He213 Hierarchical structure (dpeaa)DE-He213 Finite union of intervals (dpeaa)DE-He213 Kronecker–Weyl equidistribution along the primes (dpeaa)DE-He213 |
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misc Exponential bases misc Riesz bases misc Hierarchical structure misc Finite union of intervals misc Kronecker–Weyl equidistribution along the primes |
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misc Exponential bases misc Riesz bases misc Hierarchical structure misc Finite union of intervals misc Kronecker–Weyl equidistribution along the primes |
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A note on exponential Riesz bases |
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Sampling theory, signal processing, and data analysis |
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note on exponential riesz bases |
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A note on exponential Riesz bases |
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Abstract We prove that if %$I_\ell = [a_\ell ,b_\ell )%$, %$\ell =1,\ldots ,L%$, are disjoint intervals in [0, 1) with the property that the numbers %$1, a_1, \ldots , a_L, b_1, \ldots , b_L%$ are linearly independent over %${\mathbb {Q}}%$, then there exist pairwise disjoint sets %$\Lambda _\ell \subset {\mathbb {Z}}%$, %$\ell =1, \ldots , L%$, such that for every %$J \subset \{ 1, \ldots , L \}%$, the system %$\{e^{2\pi i \lambda x} : \lambda \in \cup _{\ell \in J} \, \Lambda _\ell \}%$ is a Riesz basis for %$L^2 ( \cup _{\ell \in J} \, I_\ell )%$. Also, we show that for any disjoint intervals %$I_\ell %$, %$\ell =1, \ldots , L%$, contained in [1, N) with %$N \in {\mathbb {N}}%$, the orthonormal basis %$\{e^{2\pi i n x} : n \in {\mathbb {Z}}\}%$ of %$L^2[0,1)%$ can be complemented by a Riesz basis %$\{e^{2\pi i \lambda x}: \lambda \in \Lambda \}%$ for %$L^2(\cup _{\ell =1}^L \, I_{\ell })%$ with some set %$\Lambda \subset (\frac{1}{N} {\mathbb {Z}}) \backslash {\mathbb {Z}}%$, in the sense that their union %$\{e^{2\pi i \lambda x} : \lambda \in {\mathbb {Z}}\cup \Lambda \}%$ is a Riesz basis for %$L^2 ( [0,1) \cup I_1 \cup \cdots \cup I_L )%$. © The Author(s) 2022 |
| abstractGer |
Abstract We prove that if %$I_\ell = [a_\ell ,b_\ell )%$, %$\ell =1,\ldots ,L%$, are disjoint intervals in [0, 1) with the property that the numbers %$1, a_1, \ldots , a_L, b_1, \ldots , b_L%$ are linearly independent over %${\mathbb {Q}}%$, then there exist pairwise disjoint sets %$\Lambda _\ell \subset {\mathbb {Z}}%$, %$\ell =1, \ldots , L%$, such that for every %$J \subset \{ 1, \ldots , L \}%$, the system %$\{e^{2\pi i \lambda x} : \lambda \in \cup _{\ell \in J} \, \Lambda _\ell \}%$ is a Riesz basis for %$L^2 ( \cup _{\ell \in J} \, I_\ell )%$. Also, we show that for any disjoint intervals %$I_\ell %$, %$\ell =1, \ldots , L%$, contained in [1, N) with %$N \in {\mathbb {N}}%$, the orthonormal basis %$\{e^{2\pi i n x} : n \in {\mathbb {Z}}\}%$ of %$L^2[0,1)%$ can be complemented by a Riesz basis %$\{e^{2\pi i \lambda x}: \lambda \in \Lambda \}%$ for %$L^2(\cup _{\ell =1}^L \, I_{\ell })%$ with some set %$\Lambda \subset (\frac{1}{N} {\mathbb {Z}}) \backslash {\mathbb {Z}}%$, in the sense that their union %$\{e^{2\pi i \lambda x} : \lambda \in {\mathbb {Z}}\cup \Lambda \}%$ is a Riesz basis for %$L^2 ( [0,1) \cup I_1 \cup \cdots \cup I_L )%$. © The Author(s) 2022 |
| abstract_unstemmed |
Abstract We prove that if %$I_\ell = [a_\ell ,b_\ell )%$, %$\ell =1,\ldots ,L%$, are disjoint intervals in [0, 1) with the property that the numbers %$1, a_1, \ldots , a_L, b_1, \ldots , b_L%$ are linearly independent over %${\mathbb {Q}}%$, then there exist pairwise disjoint sets %$\Lambda _\ell \subset {\mathbb {Z}}%$, %$\ell =1, \ldots , L%$, such that for every %$J \subset \{ 1, \ldots , L \}%$, the system %$\{e^{2\pi i \lambda x} : \lambda \in \cup _{\ell \in J} \, \Lambda _\ell \}%$ is a Riesz basis for %$L^2 ( \cup _{\ell \in J} \, I_\ell )%$. Also, we show that for any disjoint intervals %$I_\ell %$, %$\ell =1, \ldots , L%$, contained in [1, N) with %$N \in {\mathbb {N}}%$, the orthonormal basis %$\{e^{2\pi i n x} : n \in {\mathbb {Z}}\}%$ of %$L^2[0,1)%$ can be complemented by a Riesz basis %$\{e^{2\pi i \lambda x}: \lambda \in \Lambda \}%$ for %$L^2(\cup _{\ell =1}^L \, I_{\ell })%$ with some set %$\Lambda \subset (\frac{1}{N} {\mathbb {Z}}) \backslash {\mathbb {Z}}%$, in the sense that their union %$\{e^{2\pi i \lambda x} : \lambda \in {\mathbb {Z}}\cup \Lambda \}%$ is a Riesz basis for %$L^2 ( [0,1) \cup I_1 \cup \cdots \cup I_L )%$. © The Author(s) 2022 |
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| title_short |
A note on exponential Riesz bases |
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https://dx.doi.org/10.1007/s43670-022-00031-9 |
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Lee, Dae Gwan |
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10.1007/s43670-022-00031-9 |
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2024-07-03T14:29:46.231Z |
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| score |
7.402272 |