Representations by Beurling Systems
We prove that a Beurling system with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi><mo>∈</mo><msup><mi>H</mi><mi>p</mi></msup><mro...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-08-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/11/17/3663 |
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| author | Kazaros Kazarian |
| author_facet | Kazaros Kazarian |
| author_sort | Kazaros Kazarian |
| collection | DOAJ |
| description | We prove that a Beurling system with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi><mo>∈</mo><msup><mi>H</mi><mi>p</mi></msup><mrow><mo>(</mo><mi mathvariant="double-struck">D</mi><mo>)</mo></mrow><mo>,</mo><mn>1</mn><mo>≤</mo><mi>p</mi><mo><</mo><mo>∞</mo></mrow></semantics></math></inline-formula> is an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>M</mi></mrow></semantics></math></inline-formula>—basis in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>H</mi><mi>p</mi></msup><mrow><mo>(</mo><mi mathvariant="double-struck">D</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> with an explicit dual system. Any function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>∈</mo><msup><mi>H</mi><mi>p</mi></msup><mrow><mo>(</mo><mi mathvariant="double-struck">D</mi><mo>)</mo></mrow><mo>,</mo><mn>1</mn><mo>≤</mo><mi>p</mi><mo><</mo><mo>∞</mo></mrow></semantics></math></inline-formula> can be expanded as a series by the system <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mrow><mo>{</mo><msup><mi>z</mi><mi>m</mi></msup><mi>F</mi><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mo>}</mo></mrow><mrow><mi>m</mi><mo>=</mo><mn>0</mn></mrow><mo>∞</mo></msubsup><mo>.</mo></mrow></semantics></math></inline-formula> For different summation methods, we characterize the outer functions <i>F</i> for which the expansion with respect to the corresponding Beurling system converges to <i>f</i>. Related results for weighted Hardy spaces in the unit disc are studied. Particularly we prove Rosenblum’s hypothesis. |
| first_indexed | 2024-03-10T23:17:34Z |
| format | Article |
| id | doaj.art-9ab44b29101c448bbcc3483acd3a08aa |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T23:17:34Z |
| publishDate | 2023-08-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-9ab44b29101c448bbcc3483acd3a08aa2023-11-19T08:30:26ZengMDPI AGMathematics2227-73902023-08-011117366310.3390/math11173663Representations by Beurling SystemsKazaros Kazarian0Departamento de Matemáticas, Facultad de Ciencias, Universidad Autónoma de Madrid, Mod. 17, 28049 Madrid, SpainWe prove that a Beurling system with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi><mo>∈</mo><msup><mi>H</mi><mi>p</mi></msup><mrow><mo>(</mo><mi mathvariant="double-struck">D</mi><mo>)</mo></mrow><mo>,</mo><mn>1</mn><mo>≤</mo><mi>p</mi><mo><</mo><mo>∞</mo></mrow></semantics></math></inline-formula> is an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>M</mi></mrow></semantics></math></inline-formula>—basis in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>H</mi><mi>p</mi></msup><mrow><mo>(</mo><mi mathvariant="double-struck">D</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> with an explicit dual system. Any function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>∈</mo><msup><mi>H</mi><mi>p</mi></msup><mrow><mo>(</mo><mi mathvariant="double-struck">D</mi><mo>)</mo></mrow><mo>,</mo><mn>1</mn><mo>≤</mo><mi>p</mi><mo><</mo><mo>∞</mo></mrow></semantics></math></inline-formula> can be expanded as a series by the system <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mrow><mo>{</mo><msup><mi>z</mi><mi>m</mi></msup><mi>F</mi><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mo>}</mo></mrow><mrow><mi>m</mi><mo>=</mo><mn>0</mn></mrow><mo>∞</mo></msubsup><mo>.</mo></mrow></semantics></math></inline-formula> For different summation methods, we characterize the outer functions <i>F</i> for which the expansion with respect to the corresponding Beurling system converges to <i>f</i>. Related results for weighted Hardy spaces in the unit disc are studied. Particularly we prove Rosenblum’s hypothesis.https://www.mdpi.com/2227-7390/11/17/3663summation basishardy spacesouter functionBeurling systemkernelsrepresentation of functions |
| spellingShingle | Kazaros Kazarian Representations by Beurling Systems Mathematics summation basis hardy spaces outer function Beurling system kernels representation of functions |
| title | Representations by Beurling Systems |
| title_full | Representations by Beurling Systems |
| title_fullStr | Representations by Beurling Systems |
| title_full_unstemmed | Representations by Beurling Systems |
| title_short | Representations by Beurling Systems |
| title_sort | representations by beurling systems |
| topic | summation basis hardy spaces outer function Beurling system kernels representation of functions |
| url | https://www.mdpi.com/2227-7390/11/17/3663 |
| work_keys_str_mv | AT kazaroskazarian representationsbybeurlingsystems |