arXiv subjects

Eugenio Hernandez

Publications and source records attributed to Eugenio Hernandez.

5 recordsLinked to original sources

arXiv2019

Lebesgue-type inequalities in greedy approximation

We present new results regarding Lebesgue-type inequalities for the Weak Chebyshev Greedy Algorithm (WCGA) in uniformly smooth Banach spaces. We improve earlier bounds in Temlyakov (Forum Math Sigma 2014), for dictionaries satisfying a new property introduced here. We apply these results to derive optimal bounds in two natural examples of sequence spaces. In particular, optimality is obtained in the case of the multivariate Haar system in Lp with 1<p<2, under the Littlewood-Paley norm.

math.FA↗

arXiv2019

Extra invariance of principal shift invariant spaces and the Zak transform

We prove a necessary and sufficient condition for a principal shift invariant space of $L^2(\mathbb{R})$ to be invariant under translations by the subgroup $\frac{1}{N} \mathbb{Z}, N>1$. This condition is given in terms of the Zak transform of the group $\frac{1}{N} \mathbb{Z}.$ This result is extended to principal shift invariant spaces generated by a lattice in a general locally compact abelian (LCA) group.

math.CA↗

arXiv2016

Lattice sub-tilings and frames in LCA groups

Given a lattice $Λ$ in a locally compact abelian group $G$ and a measurable subset $Ω$ with finite and positive measure, then the set of characters associated to the dual lattice form a frame for $L^2(Ω)$ if and only if the distinct translates by $Λ$ of $Ω$ have almost empty intersections. Some consequences of this results are the well-known Fuglede theorem for lattices, as well as a simple characterization for frames of modulates.

math.FA↗

arXiv2016

Group Riesz and Frame Sequences: The Bracket and the Gramian

Given a discrete group and a unitary representation on a Hilbert space $\mathcal{H}$, we prove that the notions of operator Bracket map and Gramian coincide on a dense set of $\mathcal{H}$. As a consequence, combining this result with known frame theory, we can recover in a simple way all previous Bracket characterizations of Riesz and frame sequences generated by a single element under a unitary representation.

math.FA↗

arXiv2013

Bracket map for Heisenberg group and the characterization of cyclic subspaces

The bracket map was originally considered for locally compact abelian groups. In this work we extend the study of bracket maps to the noncommutative setting, providing characterizations of bases and frames for cyclic subspaces of the Heisenberg group. We also indicate how to generalize these results to a class of non-abelian nilpotent Lie groups whose irreducible representations are square integrable modulo the center.

math.FA↗