arXiv subjects

Antonio G. García

Publications and source records attributed to Antonio G. García.

10 recordsLinked to original sources

arXiv2021

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

The translation of an operator is defined by using conjugation with time-frequency shifts. Thus, one can define $Λ$-shift-invariant subspaces of Hilbert-Schmidt operators, finitely generated, with respect to a lattice $Λ$ 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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arXiv2021

Average sampling in certain subspaces of Hilbert-Schmidt operators on $L^2(\mathbb{R}^d)$

The concept of translation of an operator allows to consider the analogous of shift-invariant subspaces in the class of Hilbert-Schmidt operators. Thus, we extend the concept of average sampling to this new setting, and we obtain the corresponding sampling formulas. The key point here is the use of the Weyl transform, a unitary mapping between the space of square integrable functions in the phase space $\mathbb{R}^d\times \widehat{\mathbb{R}}^d$ and the Hilbert space of Hilbert-Schmidt operators on $L^2(\mathbb{R}^d)$, which permits to take advantage of some well established sampling results.

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arXiv2020

Sampling in the range of the analysis operator of a continuous frame having unitary structure

We establish a regular sampling theory in the range of the analysis operator of a continuous frame having a unitary structure. The unitary structure is related with a unitary representation of a locally compact abelian group on a separable Hilbert space. The samples are defined by means of suitable discrete convolution systems which generalize some usual sampling settings; here regular sampling means that the samples are taken at a countable discrete subgroup.

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arXiv2020

Convolution systems on discrete abelian groups as a unifying strategy in sampling theory

A regular sampling theory in a multiply generated unitary invariant subspace of a separable Hilbert space $\mathcal{H}$ is proposed. This subspace is associated to a unitary representation of a countable discrete abelian group $G$ on $\mathcal{H}$. The samples are defined by means of a filtering process which generalizes the usual sampling settings. The multiply generated setting allows to consider some examples where the group $G$ is non-abelian as, for instance, crystallographic groups. Finally, it is worth to mention that classical average or pointwise sampling in shift-invariant subspaces are particular examples included in the followed approach.

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arXiv2018

Knit product of finite groups and sampling

A finite sampling theory associated with a unitary representation of a finite non Abelian group $\mathbf{G}$ on a Hilbert space is stablished. The non Abelian group $\mathbf{G}$ is a knit product $\mathbf{N}\bowtie \mathbf{H}$ of two finite subgroups $\mathbf{N}$ and $\mathbf{H}$. Sampling formulas where the samples are indexed by either $\mathbf{N}$ or $\mathbf{H}$ are obtained. Using suitable expressions for the involved samples, the problem is reduced to obtain dual frames in the Hilbert space $\ell^2(\mathbf{G})$ having a unitary invariance property; this is done by using matrix analysis techniques. An example involving dihedral groups illustrates the obtained sampling results.

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arXiv2018

Semi-direct product of groups, filter banks and sampling

An abstract sampling theory associated to a unitary representation of a countable discrete non abelian group $G$, which is a semi-direct product of groups, on a separable Hilbert space is studied. A suitable expression of the data samples and the use of a filter bank formalism allows to fix the mathematical problem to be solved: the search of appropriate dual frames for $\ell^2(G)$. An example involving crystallographic groups illustrates the obtained results by using average or pointwise samples.

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arXiv2018

Regular generalized sampling in T-invariant subspaces of a Hilbert space

A regular generalized sampling theory in some structured T-invariant subspaces of a Hilbert space H, where T denotes a bounded invertible operator in H, is established in this paper. This is done by walking through the most important cases which generalize the usual sampling settings.

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arXiv2017

Sampling formulas involving differences in shift-invariant subspaces: a unified approach

Successive differences on a sequence of data help to discover some smoothness features of this data. This was one of the main reasons for rewriting the classical interpolation formula in terms of such data differences. The aim of this paper is to mimic them to a sequence of regular samples of a function in a shift-invariant subspace allowing its stable recovery. A suitable expression for the functions in the shift-invariant subspace by means of an isomorphism with the $L^2(0,1)$ space is the key to identify the simple pattern followed by the dual Riesz bases involved in the derived formulas. The paper contains examples illustrating different non-exhaustive situations including also the two-dimensional case.

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arXiv2016

Modeling sampling in tensor products of unitary invariant subspaces

The use of unitary invariant subspaces of a Hilbert space $\mathcal{H}$ is nowadays a recognized fact in the treatment of sampling problems. Indeed, shift-invariant subspaces of $L^2(\mathbb{R})$ and also periodic extensions of finite signals are remarkable examples where this occurs. As a consequence, the availability of an abstract unitary sampling theory becomes a useful tool to handle these problems. In this paper we derive a sampling theory for tensor products of unitary invariant subspaces. This allows to merge the cases of finitely/infinitely generated unitary invariant subspaces formerly studied in the mathematical literature, it also allows to introduce the several variables case. As the involved samples are identified as frame coefficients in suitable tensor product spaces, the relevant mathematical technique is that of frame theory, involving both, finite/infinite dimensional cases.

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arXiv2015

Towards a quantum sampling theory: the case of finite groups

Nyquist-Shannon sampling theorem, instrumental in classical telecommunication technologies, is extended to quantum systems supporting a unitary representation of a finite group $G$. Two main ideas from the classical theory having natural counterparts in the quantum setting: frames and invariant subspaces, provide the mathematical background for the theory. The main ingredients of classical sampling theorems are discussed and their quantum counterparts are thoroughly analyzed in this simple situation. A few examples illustrating the obtained results are discussed.

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