Scalability of Frames Generated by Dynamical Operators
Let ℍ be a separable Hilbert space, let G ⊂ ℍ, and let A be an operator on ℍ. Under appropriate conditions on A and G, it is known that the set of iterations FG(A)={Ajg|g∈G,0≤j≤L(g)} is a frame for ℍ. We call FG(A) a dynamical frame for ℍ, and explore further its properties; in particular, we show t...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Frontiers Media S.A.
2017-11-01
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| Series: | Frontiers in Applied Mathematics and Statistics |
| Subjects: | |
| Online Access: | http://journal.frontiersin.org/article/10.3389/fams.2017.00022/full |
| _version_ | 1818107438288601088 |
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| author | Roza Aceska Yeon H. Kim |
| author_facet | Roza Aceska Yeon H. Kim |
| author_sort | Roza Aceska |
| collection | DOAJ |
| description | Let ℍ be a separable Hilbert space, let G ⊂ ℍ, and let A be an operator on ℍ. Under appropriate conditions on A and G, it is known that the set of iterations FG(A)={Ajg|g∈G,0≤j≤L(g)} is a frame for ℍ. We call FG(A) a dynamical frame for ℍ, and explore further its properties; in particular, we show that the canonical dual frame of FG(A) also has an iterative set structure. We explore the relations between the operator A, the set G and the number of iterations L which ensure that the system FG(A) is a scalable frame. We give a general statement on frame scalability, and study in detail the case when A is a normal operator, utilizing the unitary diagonalization. In addition, we answer the question of when FG(A) is a scalable frame in several special cases involving block-diagonal and companion operators. |
| first_indexed | 2024-12-11T01:59:28Z |
| format | Article |
| id | doaj.art-47553859b1c94bb6bfbbccbc8b23ee78 |
| institution | Directory Open Access Journal |
| issn | 2297-4687 |
| language | English |
| last_indexed | 2024-12-11T01:59:28Z |
| publishDate | 2017-11-01 |
| publisher | Frontiers Media S.A. |
| record_format | Article |
| series | Frontiers in Applied Mathematics and Statistics |
| spelling | doaj.art-47553859b1c94bb6bfbbccbc8b23ee782022-12-22T01:24:31ZengFrontiers Media S.A.Frontiers in Applied Mathematics and Statistics2297-46872017-11-01310.3389/fams.2017.00022307634Scalability of Frames Generated by Dynamical OperatorsRoza Aceska0Yeon H. Kim1Department of Mathematical Sciences, Ball State University, Muncie, IN, United StatesDepartment of Mathematics, Central Michigan University, Mount Pleasant, MI, United StatesLet ℍ be a separable Hilbert space, let G ⊂ ℍ, and let A be an operator on ℍ. Under appropriate conditions on A and G, it is known that the set of iterations FG(A)={Ajg|g∈G,0≤j≤L(g)} is a frame for ℍ. We call FG(A) a dynamical frame for ℍ, and explore further its properties; in particular, we show that the canonical dual frame of FG(A) also has an iterative set structure. We explore the relations between the operator A, the set G and the number of iterations L which ensure that the system FG(A) is a scalable frame. We give a general statement on frame scalability, and study in detail the case when A is a normal operator, utilizing the unitary diagonalization. In addition, we answer the question of when FG(A) is a scalable frame in several special cases involving block-diagonal and companion operators.http://journal.frontiersin.org/article/10.3389/fams.2017.00022/fulldynamical samplingframesscalable framesiterative actions of operators |
| spellingShingle | Roza Aceska Yeon H. Kim Scalability of Frames Generated by Dynamical Operators Frontiers in Applied Mathematics and Statistics dynamical sampling frames scalable frames iterative actions of operators |
| title | Scalability of Frames Generated by Dynamical Operators |
| title_full | Scalability of Frames Generated by Dynamical Operators |
| title_fullStr | Scalability of Frames Generated by Dynamical Operators |
| title_full_unstemmed | Scalability of Frames Generated by Dynamical Operators |
| title_short | Scalability of Frames Generated by Dynamical Operators |
| title_sort | scalability of frames generated by dynamical operators |
| topic | dynamical sampling frames scalable frames iterative actions of operators |
| url | http://journal.frontiersin.org/article/10.3389/fams.2017.00022/full |
| work_keys_str_mv | AT rozaaceska scalabilityofframesgeneratedbydynamicaloperators AT yeonhkim scalabilityofframesgeneratedbydynamicaloperators |