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

Antonio G. García

doi:10.3934/mfc.2021019

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2021, Volume 4, Issue 4: 281-297. Doi: 10.3934/mfc.2021019

This issue Previous Article On multidimensional Urysohn type generalized sampling operators Next Article A note on convergence results for varying interval valued multisubmeasures

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

Antonio G. García

Departamento de Matemáticas, Universidad Carlos Ⅲ de Madrid, Leganés, 28911, Spain

Received: March 31, 2021

Revised: July 31, 2021

Early access: September 2021

Published: November 2021

The author is supported by the grant MTM2017-84098-P from the Spanish Ministerio de Economía y Competitividad (MINECO)

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Abstract

The translation of an operator is defined by using conjugation with time-frequency shifts. Thus, one can define $ \Lambda $-shift-invariant subspaces of Hilbert-Schmidt operators, finitely generated, with respect to a lattice $ \Lambda $ in $ \mathbb{R}^{2d} $. 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.

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Mathematics Subject Classification: Primary: 42C15, 43A32, 47B10 Secondary: 94A20.

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