Biframes and some of their properties
Abstract Recently, frame multipliers, pair frames, and controlled frames have been investigated to improve the numerical efficiency of iterative algorithms for inverting the frame operator and other applications of frames. In this paper, the concept of biframe is introduced for a Hilbert space. A bi...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2022-08-01
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| Series: | Journal of Inequalities and Applications |
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| Online Access: | https://doi.org/10.1186/s13660-022-02844-7 |
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| author | Maryam Firouzi Parizi Azadeh Alijani Mohammad Ali Dehghan |
| author_facet | Maryam Firouzi Parizi Azadeh Alijani Mohammad Ali Dehghan |
| author_sort | Maryam Firouzi Parizi |
| collection | DOAJ |
| description | Abstract Recently, frame multipliers, pair frames, and controlled frames have been investigated to improve the numerical efficiency of iterative algorithms for inverting the frame operator and other applications of frames. In this paper, the concept of biframe is introduced for a Hilbert space. A biframe is a pair of sequences in a Hilbert space that applies to an inequality similar to a frame inequality. Also, it can be regarded as a generalization of controlled frames and a special kind of pair frames. The basic properties of biframes are investigated based on the biframe operator. Then, biframes are classified based on the type of their constituent sequences. In particular, biframes for which one of the constituent sequences is an orthonormal basis { e k } k = 1 ∞ $\{e_{k}\}_{k=1}^{\infty}$ are studied. Then, a new class of Riesz bases denoted by [ { e k } ] $[\{e_{k}\}]$ is introduced and is called b-Riesz bases. An interesting result is also proved, showing that the set of all b-Riesz bases is a proper subset of the set of all Riesz bases. More precisely, b-Riesz bases induce an equivalence relation on [ { e k } ] $[\{e_{k}\}]$ . |
| format | Article |
| id | doaj-art-64b86d0e39ab4c168d39921903e0a6e3 |
| institution | Kabale University |
| issn | 1029-242X |
| language | English |
| publishDate | 2022-08-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of Inequalities and Applications |
| spelling | doaj-art-64b86d0e39ab4c168d39921903e0a6e32025-02-02T12:47:23ZengSpringerOpenJournal of Inequalities and Applications1029-242X2022-08-012022112410.1186/s13660-022-02844-7Biframes and some of their propertiesMaryam Firouzi Parizi0Azadeh Alijani1Mohammad Ali Dehghan2Department of Mathematics, Vali-e-Asr UniversityDepartment of Mathematics, Vali-e-Asr UniversityDepartment of Mathematics, Vali-e-Asr UniversityAbstract Recently, frame multipliers, pair frames, and controlled frames have been investigated to improve the numerical efficiency of iterative algorithms for inverting the frame operator and other applications of frames. In this paper, the concept of biframe is introduced for a Hilbert space. A biframe is a pair of sequences in a Hilbert space that applies to an inequality similar to a frame inequality. Also, it can be regarded as a generalization of controlled frames and a special kind of pair frames. The basic properties of biframes are investigated based on the biframe operator. Then, biframes are classified based on the type of their constituent sequences. In particular, biframes for which one of the constituent sequences is an orthonormal basis { e k } k = 1 ∞ $\{e_{k}\}_{k=1}^{\infty}$ are studied. Then, a new class of Riesz bases denoted by [ { e k } ] $[\{e_{k}\}]$ is introduced and is called b-Riesz bases. An interesting result is also proved, showing that the set of all b-Riesz bases is a proper subset of the set of all Riesz bases. More precisely, b-Riesz bases induce an equivalence relation on [ { e k } ] $[\{e_{k}\}]$ .https://doi.org/10.1186/s13660-022-02844-7BiframesB-Riesz basesControlled framesFramesMultiplier operatorsPair frames |
| spellingShingle | Maryam Firouzi Parizi Azadeh Alijani Mohammad Ali Dehghan Biframes and some of their properties Journal of Inequalities and Applications Biframes B-Riesz bases Controlled frames Frames Multiplier operators Pair frames |
| title | Biframes and some of their properties |
| title_full | Biframes and some of their properties |
| title_fullStr | Biframes and some of their properties |
| title_full_unstemmed | Biframes and some of their properties |
| title_short | Biframes and some of their properties |
| title_sort | biframes and some of their properties |
| topic | Biframes B-Riesz bases Controlled frames Frames Multiplier operators Pair frames |
| url | https://doi.org/10.1186/s13660-022-02844-7 |
| work_keys_str_mv | AT maryamfirouziparizi biframesandsomeoftheirproperties AT azadehalijani biframesandsomeoftheirproperties AT mohammadalidehghan biframesandsomeoftheirproperties |