Operators commuting with a discrete subgroup of translations

Feichtinger, Hans G.; Führ, Hartmut; Gröchenig, Karlheinz; Kaiblinger, Norbert

doi:10.1007/bf02930987

Cited by 2 publications

(2 citation statements)

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“…By Proposition 5.2 , one obtains a characterization of the bounded bijective operators on

which commute with the operators of translates on a discrete subgroup. For another structure theorem for operators from the Schwartz space

into the tempered distributions

, which commute with a discrete subgroup of translations, we refer to [14] .…”

Section: Multipliers For Pseudo-coherent Framesmentioning

confidence: 99%

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Representation of the inverse of a frame multiplier

Balázs

Stoeva

2015

Journal of Mathematical Analysis and Applications

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Certain mathematical objects appear in a lot of scientific disciplines, like physics, signal processing and, naturally, mathematics. In a general setting they can be described as frame multipliers, consisting of analysis, multiplication by a fixed sequence (called the symbol), and synthesis. In this paper we show a surprising result about the inverse of such operators, if any, as well as new results about a core concept of frame theory, dual frames. We show that for semi-normalized symbols, the inverse of any invertible frame multiplier can always be represented as a frame multiplier with the reciprocal symbol and dual frames of the given ones. Furthermore, one of those dual frames is uniquely determined and the other one can be arbitrarily chosen. We investigate sufficient conditions for the special case, when both dual frames can be chosen to be the canonical duals. In connection to the above, we show that the set of dual frames determines a frame uniquely. Furthermore, for a given frame, the union of all coefficients of its dual frames is dense in ℓ2. We also introduce a class of frames (called pseudo-coherent frames), which includes Gabor frames and coherent frames, and investigate invertible pseudo-coherent frame multipliers, allowing a classification for frame-type operators for these frames. Finally, we give a numerical example for the invertibility of multipliers in the Gabor case.

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“…By Proposition 5.2 , one obtains a characterization of the bounded bijective operators on

which commute with the operators of translates on a discrete subgroup. For another structure theorem for operators from the Schwartz space

into the tempered distributions

, which commute with a discrete subgroup of translations, we refer to [14] .…”

Section: Multipliers For Pseudo-coherent Framesmentioning

confidence: 99%

“…Note that frame multipliers with a constant symbol are the so-called frame-type operators (see, e.g., [16,14,35] ) or mixed frame operators (see, e.g., [11] ), as the frame multiplier

corresponds to the frame-type operator denoted as

and for