Frames and generalized operator orbits
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Published:2023-07-06
Issue:2
Volume:21
Page:
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ISSN:2730-5716
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Container-title:Sampling Theory, Signal Processing, and Data Analysis
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language:en
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Short-container-title:Sampl. Theory Signal Process. Data Anal.
Author:
Christensen Ole,Hasannasab Marzieh
Abstract
AbstractThis short note is concerned with various operator representations of frames in a Hilbert space $${{{\mathcal {H}}}}.$$
H
.
While it is known that only very special frames can be represented in the form $$\{ T^n \varphi \}_{n=0}^\infty $$
{
T
n
φ
}
n
=
0
∞
for a bounded operator T, we prove that every frame (actually, every Bessel sequence) has a representation of the form $$\{UT^k \varphi \}_{k=0}^\infty $$
{
U
T
k
φ
}
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 $$\{PT^k \varphi \}_{k=0}^\infty ,$$
{
P
T
k
φ
}
k
=
0
∞
,
where T is a bounded operator on an ambient Hilbert space and P denotes the orthogonal projection onto the given Hilbert space $${{{\mathcal {H}}}}.$$
H
.
In particular, this implies that for any frame, the frame coefficients of $$f\in {{{\mathcal {H}}}}$$
f
∈
H
can be calculated as inner products between f and a system of functions of the form $$\{T^k \varphi \}_{k=0}^\infty ,$$
{
T
k
φ
}
k
=
0
∞
,
for a bounded operator T on an ambient Hilbert space.
Funder
Universität zu Lübeck
Publisher
Springer Science and Business Media LLC
Subject
Computational Mathematics,Radiology, Nuclear Medicine and imaging,Signal Processing,Algebra and Number Theory,Analysis
Reference8 articles.
1. Aldroubi, A., Cabrelli, C., Molter, U., Tang, S.: Dynamical sampling. Appl. Harmon. Anal. Appl. 42(3), 378–401 (2017) 2. Aldroubi, A., Cabrelli, C., Çakmak, A.F., Molter, U., Petrosyan, A.: Iterative actions of normal operators. J. Funct. Anal. 272(3), 1121–1146 (2017) 3. Aldroubi, A., Petrosyan, A.: Dynamical sampling and systems from iterative actions of operators. In: Mhaskar, H., Pesenson, I., Zhou, D.X., Le Gia, Q.T., Mayeli, A. (eds.) Frames and Other Bases in Abstract and Function Spaces. Birkhäuser, Boston (2017) 4. Christensen, O., Hasannasab, M.: Frame properties of systems arising via iterative actions of operators. Appl. Comput. Harmon. Anal. 46, 664–673 (2019) 5. Christensen, O., Hasannasab, M.: Operator representations of frames: boundedness, duality, and stability. Integr. Equ. Oper. Theory 88(4), 483–499 (2017)
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