2,271 research outputs found
Frames of subspaces and operators
We study the relationship between operators, orthonormal basis of subspaces
and frames of subspaces (also called fusion frames) for a separable Hilbert
space . We get sufficient conditions on an orthonormal basis of
subspaces of a Hilbert space
and a surjective in order that
is a frame of subspaces with respect to a computable
sequence of weights. We also obtain generalizations of results in [J. A.
Antezana, G. Corach, M. Ruiz and D. Stojanoff, Oblique projections and frames.
Proc. Amer. Math. Soc. 134 (2006), 1031-1037], which related frames of
subspaces (including the computation of their weights) and oblique projections.
The notion of refinament of a fusion frame is defined and used to obtain
results about the excess of such frames. We study the set of admissible weights
for a generating sequence of subspaces. Several examples are given.Comment: 21 pages, LaTeX; added references and comments about fusion frame
Robust dual reconstruction systems and fusion frames
We study the duality of reconstruction systems, which are g-frames in a finite dimensional setting. These systems allow redundant linear encoding-decoding schemes implemented by the so-called dual reconstruction systems. We are particularly interested in the projective reconstruction systems that are the analogue of fusion frames in this context. Thus, we focus on dual systems of a fixed projective system that are optimal with respect to erasures of the reconstruction system coefficients involved in the decoding process. We consider two different measures of the reconstruction error in a blind reconstruction algorithm. We also study the projective reconstruction system that best approximate an arbitrary reconstruction system, based on some well known results in matrix theory. Finally, we present a family of examples in which the problem of existence of a dual projective system of a reconstruction system of this type is considered.Fil: Massey, Pedro Gustavo. Consejo Nacional de Investigaciones CientÃficas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; ArgentinaFil: Ruiz, Mariano Andres. Consejo Nacional de Investigaciones CientÃficas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; ArgentinaFil: Stojanoff, Demetrio. Consejo Nacional de Investigaciones CientÃficas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentin
Multiplicative Lidskii's inequalities and optimal perturbations of frames
In this paper we study two design problems in frame theory: on the one hand,
given a fixed finite frame \cF for \hil\cong\C^d we compute those dual
frames \cG of \cF that are optimal perturbations of the canonical dual
frame for \cF under certain restrictions on the norms of the elements of
\cG. On the other hand, for a fixed finite frame \cF=\{f_j\}_{j\in\In} for
\hil we compute those invertible operators such that is a
perturbation of the identity and such that the frame V\cdot
\cF=\{V\,f_j\}_{j\in\In} - which is equivalent to \cF - is optimal among
such perturbations of \cF. In both cases, optimality is measured with respect
to submajorization of the eigenvalues of the frame operators. Hence, our
optimal designs are minimizers of a family of convex potentials that include
the frame potential and the mean squared error. The key tool for these results
is a multiplicative analogue of Lidskii's inequality in terms of
log-majorization and a characterization of the case of equality.Comment: 22 page
Optimal dual frames and frame completions for majorization
In this paper we consider two problems in frame theory. On the one hand,
given a set of vectors we describe the spectral and geometrical
structure of optimal completions of by a finite family of vectors
with prescribed norms, where optimality is measured with respect to
majorization. In particular, these optimal completions are the minimizers of a
family of convex functionals that include the mean square error and the
Bendetto-Fickus' frame potential. On the other hand, given a fixed frame
we describe explicitly the spectral and geometrical structure of
optimal frames that are in duality with and such that
the Frobenius norms of their analysis operators is bounded from below by a
fixed constant. In this case, optimality is measured with respect to
submajorization of the frames operators. Our approach relies on the description
of the spectral and geometrical structure of matrices that minimize
submajorization on sets that are naturally associated with the problems above.Comment: 29 pages, with modifications related with the exposition of the
materia
Duality in reconstruction systems
We consider reconstruction systems (RS's), which are G-frames in a finite dimensional setting, and that includes the fusion frames as projective RS's. We describe the spectral picture of the set of RS operators for the projective systems with fixed weights. We also introduce a functional defined on dual pairs of RS's, called the joint potential, and study the structure of the minimizers of this functional. In the case of irreducible RS's the minimizers are characterize as the tight systems. In the general case we give spectral and geometric characterizations of the minimizers of the joint potential. At the end of the paper we show several examples that illustrate our results.Fil: Massey, Pedro Gustavo. Consejo Nacional de Investigaciones CientÃficas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentina. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Departamento de Matemáticas; ArgentinaFil: Ruiz, Mariano Andres. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Departamento de Matemáticas; Argentina. Consejo Nacional de Investigaciones CientÃficas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; ArgentinaFil: Stojanoff, Demetrio. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Departamento de Matemáticas; Argentina. Consejo Nacional de Investigaciones CientÃficas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentin
Weighted projections and Riesz frames
Let be a (separable) Hilbert space and a
fixed orthonormal basis of . Motivated by many papers on scaled
projections, angles of subspaces and oblique projections, we define and study
the notion of compatibility between a subspace and the abelian algebra of
diagonal operators in the given basis. This is used to refine previous work on
scaled projections, and to obtain a new characterization of Riesz frames.Comment: 23 pages, to appear in Linear Algebra and its Application
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