Quantum Harmonic Analysis on Lattices and Gabor Multipliers

Skrettingland, Eirik

doi:10.1007/s00041-020-09759-1

Cited by 16 publications

(24 citation statements)

References 51 publications

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“…Consequently, as it was pointed out in Refs. [6,24], for fixed S ∈ HS(R d ) with Weyl symbol a S ∈ L 2 (R 2d ) and lattice Λ in R 2d , the sequence {α λ (S)} λ∈Λ is a Riesz sequence in HS(R d ), i.e., a Riesz basis for V 2 S := span HS α λ (S) λ∈Λ , if and only if the sequence {T λ (a S )} λ∈Λ is a Riesz sequence in L 2 (R 2d ), i.e., a Riesz basis for the shift-invariant subspace V 2 a S in L 2 (R 2d ) generated by a S .…”

Section: 3mentioning

confidence: 99%

“…A necessary and sufficient condition for {α λ (S)} λ∈Λ to be a Riesz sequence in HS(R d ) is given in Ref. [24]. There, it is assumed that S ∈ B, a Banach space of continuous operators with Weyl symbol a S in the Feichtinger's algebra S 0 (R 2d ); in essence, B consists of trace class operators on L 2 (R d ) with a norm-continuous inclusion ι : B → T 1 (see the details in Refs.…”

Section: 3mentioning

confidence: 99%

“…With this norm, S 0 (R d ) is a Banach space of continuous functions and an algebra under multiplication and convolution; see the details in Refs. [16,19,24].…”

Section: 3mentioning

confidence: 99%

“…More information and details about these transforms, also valid in more general settings, can be found in Refs. [7,9,16,24,26].…”

mentioning

confidence: 99%

“…4) the corresponding dualities S(R d ), S ′ (R d ) and S(R 2d ), S ′ (R 2d ) ; see, for instance, Refs. [16,24].…”

mentioning

confidence: 99%

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Sampling in $ \Lambda $-shift-invariant subspaces of Hilbert-Schmidt operators on $ L^2(\mathbb{R}^d) $

Garcı́a¹

2021

MFC

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<p style='text-indent:20px;'>The translation of an operator is defined by using conjugation with time-frequency shifts. Thus, one can define <inline-formula><tex-math id="M3">\begin{document}$ \Lambda $\end{document}</tex-math></inline-formula>-shift-invariant subspaces of Hilbert-Schmidt operators, finitely generated, with respect to a lattice <inline-formula><tex-math id="M4">\begin{document}$ \Lambda $\end{document}</tex-math></inline-formula> in <inline-formula><tex-math id="M5">\begin{document}$ \mathbb{R}^{2d} $\end{document}</tex-math></inline-formula>. These spaces can be seen as a generalization of classical shift-invariant subspaces of square integrable functions. Obtaining sampling results for these subspaces appears as a natural question that can be motivated by the problem of channel estimation in wireless communications. These sampling results are obtained in the light of the frame theory in a separable Hilbert space.</p>

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Section: 3mentioning

confidence: 99%

Section: 3mentioning

confidence: 99%

“…With this norm, S 0 (R d ) is a Banach space of continuous functions and an algebra under multiplication and convolution; see the details in Refs. [16,19,24].…”

Section: 3mentioning

confidence: 99%

“…More information and details about these transforms, also valid in more general settings, can be found in Refs. [7,9,16,24,26].…”

mentioning

confidence: 99%

“…4) the corresponding dualities S(R d ), S ′ (R d ) and S(R 2d ), S ′ (R 2d ) ; see, for instance, Refs. [16,24].…”

mentioning

confidence: 99%

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