122 research outputs found
Singular value estimates of oblique projections
Let W and M be two finite dimensional subspaces of a Hilbert space H such that H = W ⊕ M⊥, and let PW {norm of matrix} M⊥ denote the oblique projection with range W and nullspace M⊥. In this article we get the following formula for the singular values of PW {norm of matrix} M⊥2 (sk (PW {norm of matrix} M⊥) - 1) = under(min, (F, H) ∈ X (W, M))2,where the minimum is taken over the set of all operator pairs (F, H) on H such that R (F) = W, R (H) = M and FH* = PW {norm of matrix} M⊥, and k ∈ {1, ..., dim W}. We also characterize all the pairs where the minimum is attained.Fil: Antezana, Jorge Abel. 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: Corach, Gustavo. Consejo Nacional de Investigaciones CientÃficas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentina. Universidad de Buenos Aires; Argentin
Sampling theory, oblique projections and a question by Smale and Zhou
In a recent article, Smale and Zhou define a notion of rich data for sampling problems and reconstruction of signals from a discrete set of samples and study different least-square problems related with the minimization of the error. They obtain different error estimations assuming that the original signal belong to the reconstruction subspace and they propose to find error estimations if this assumption does not hold. In this paper, using projection methods, we find such estimates and we extend from reproducing kernel Hilbert spaces to abstract Hilbert spaces some of their results on function reconstruction from point values.Fil: Antezana, Jorge Abel. Consejo Nacional de Investigaciones CientÃficas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; ArgentinaFil: Corach, Gustavo. Consejo Nacional de Investigaciones CientÃficas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentin
Weighted projections into closed subspaces
In this paper we study -projections, i.e. operators of a Hilbert space
\HH which act as projections when a seminorm is considered in \HH.
-projections were introduced by Mitra and Rao \cite{[MitRao74]} for finite
dimensional spaces. We relate this concept to the theory of compatibility
between positive operators and closed subspaces of \HH. We also study the
relationship between weighted least squares problems and compatibility
Projections in operator ranges
If \H is a Hilbert space, is a positive bounded linear operator on
\cH and \cS is a closed subspace of \cH, the relative position between
\cS and A^{-1}(\cS \orto) establishes a notion of compatibility. We show
that the compatibility of (A,\cS) is equivalent to the existence of a
convenient orthogonal projection in the operator range with its
canonical Hilbertian structure
Bilateral Shorted Operators and Parallel Sums
In this paper we study shorted operators relative to two different subspaces,
for bounded operators on infinite dimensional Hilbert spaces. We define two
notions of complementability in the sense of Ando for operators, and study the
properties of the shorted operators when they can be defined. We use these
facts in order to define and study the notions of parallel sum and
substraction, in this Hilbertian context
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