Abstract
This short note is concerned with various operator representations of frames in a Hilbert space H. While it is known that only very special frames can be represented in the form {Tnφ}∞/n=0 for a bounded operator T, we prove that every frame (actually, every Bessel sequence) has a representation of the form {UTkφ}∞/k=0 for certain bounded operators U and T. We also provide a lifting procedure that allows to represent any given Bessel sequence in the form {PTkφ}∞/k=0, where T is a bounded operator on an ambient Hilbert space and P denotes the orthogonal projection onto the given Hilbert space H. In particular, this implies that for any frame, the frame coefficients of f∈ H can be calculated as inner products between ƒ and a system of functions of the form {Tkφ}∞/k=0, for a bounded operator T on an ambient Hilbert space.
Original language | English |
---|---|
Article number | 22 |
Journal | Sampling Theory, Signal Processing, and Data Analysis |
Volume | 21 |
Number of pages | 7 |
ISSN | 2730-5716 |
DOIs | |
Publication status | Published - 2023 |
Keywords
- Approximate dual frames
- Naimark’s Theorem
- Operator representation of frames