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Frame properties of operator orbits
Author(s) -
Christensen Ole,
Hasannasab Marzieh,
Philipp Friedrich
Publication year - 2020
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201800344
Subject(s) - mathematics , bounded function , operator (biology) , bounded operator , hilbert space , frame (networking) , gabor wavelet , open set , pure mathematics , set (abstract data type) , wavelet , discrete mathematics , mathematical analysis , wavelet transform , artificial intelligence , computer science , discrete wavelet transform , telecommunications , biochemistry , chemistry , repressor , transcription factor , gene , programming language
We consider sequences in a Hilbert space H of the form( T n f 0 ) n ∈ I , with a linear operator T , the index set being either I = N or I = Z , a vectorf 0 ∈ H , and answer the following two related questions: (a) Which frames for H are of this form with an at least closable operator T ? and (b) For which bounded operators T and vectors f 0 is( T n f 0 ) n ∈ Ia frame for H ? As a consequence of our results, it turns out that an overcomplete Gabor or wavelet frame can never be written in the form( T n f 0 ) n ∈ Nwith a bounded operator T . The corresponding problem for I = Z remains open. Despite the negative result for Gabor and wavelet frames, the results demonstrate that the class of frames that can be represented in the form( T n f 0 ) n ∈ Nwith a bounded operator T is significantly larger than what could be expected from the examples known so far.