Applied Mathematics to Control, Systems and Signals Research Grouphttp://hdl.handle.net/10016/61982019-09-21T13:53:45Z2019-09-21T13:53:45ZSampling Associated with a Unitary Representation of a Semi-Direct Product of Groups: A Filter Bank ApproachGarcía García, AntonioHernández Medina, Miguel ÁngelPérez Villalón, Gerardohttp://hdl.handle.net/10016/285502019-07-06T00:03:06Z2019-04-12T00:00:00ZSampling Associated with a Unitary Representation of a Semi-Direct Product of Groups: A Filter Bank Approach
García García, Antonio; Hernández Medina, Miguel Ángel; Pérez Villalón, Gerardo
An abstract sampling theory associated with 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, the use of a filter bank formalism and the corresponding frame analysis allow for fixing the mathematical problem to be solved: the search of appropriate dual frames for l2(G). An example involving crystallographic groups illustrates the obtained results by using either average or pointwise samples.
This article belongs to the Special Issue New trends on Symmetry and Topology in Quantum Mechanics
2019-04-12T00:00:00ZOn Some Sampling-Related Frames in U-Invariant SpacesFernández Morales, Héctor RaúlGarcía García, AntonioHernández-Medina, M. A.Muñoz-Bouzo, María Joséhttp://hdl.handle.net/10016/180462019-05-09T13:28:16Z2013-10-01T00:00:00ZOn Some Sampling-Related Frames in U-Invariant Spaces
Fernández Morales, Héctor Raúl; García García, Antonio; Hernández-Medina, M. A.; Muñoz-Bouzo, María José
This paper is concerned with the characterization as frames of some sequences in -invariant spaces of a separable Hilbert space H where U denotes an unitary operator defined on H ; besides, the dual frames having the same form are also found. This general setting includes, in particular, shift-invariant or modulation-invariant subspaces in L2(R) , where these frames are intimately related to the generalized sampling problem. We also deal with some related perturbation problems. In doing so, we need the unitary operator to belong to a continuous group of unitary operators.
2013-10-01T00:00:00ZOn the tomographic picture of quantum mechanicsIbort, AlbertoMan'ko, V. I.Marmo, G.Simoni, A.Ventriglia, F.http://hdl.handle.net/10016/86912019-05-09T13:28:15Z2010-06-07T00:00:00ZOn the tomographic picture of quantum mechanics
Ibort, Alberto; Man'ko, V. I.; Marmo, G.; Simoni, A.; Ventriglia, F.
We formulate necessary and sufficient conditions for a symplectic tomogram of a quantum state to determine the density state. We establish a connection between the (re)construction by means of symplectic tomograms with the construction by means of Naimark positive definite functions on the Weyl-Heisenberg group. This connection is used to formulate properties which guarantee that tomographic probabilities describe quantum states in the probability representation of quantum mechanics.
4 pages, no figures.-- PACS codes: 03.65 Sq; 03.65.Wj.-- ArXiv pre-print available at: http://arxiv.org/abs/1004.0102
2010-06-07T00:00:00ZOversampling in shift-invariant spaces with a rational sampling periodGarcía, Antonio G.Hernández-Medina, M. A.Pérez-Villalón, G.http://hdl.handle.net/10016/63162019-05-09T13:28:14Z2009-09-01T00:00:00ZOversampling in shift-invariant spaces with a rational sampling period
García, Antonio G.; Hernández-Medina, M. A.; Pérez-Villalón, G.
It is well known that, under appropriate hypotheses, a sampling formula allows us to recover any function in a principal shift-invariant space from its samples taken with sampling period one. Whenever the generator of the shift-invariant space satisfies the Strang-Fix conditions of order r, this formula also provides an approximation scheme of order r valid for smooth functions. In this paper we obtain sampling formulas sharing the same features by using a rational sampling period less than one. With the use of this oversampling technique, there is not one but an infinite number of sampling formulas. Whenever the generator has compact support, among these formulas it is possible to find one whose associated reconstruction functions have also compact support.
8 pages, no figures.
2009-09-01T00:00:00Z