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2010 Redundant decompositions, angles between subspaces and oblique projections
G. Corach, A. Maestripieri
Publ. Mat. 54(2): 461-484 (2010).

Abstract

Let ${\mathcal H}$ be a complex Hilbert space. We study the relationships between the angles between closed subspaces of ${\mathcal H}$, the oblique projections associated to non direct decompositions of ${\mathcal H}$ and a notion of compatibility between a positive (semidefinite) operator $A$ acting on ${\mathcal H}$ and a closed subspace ${\mathcal S}$ of ${\mathcal H}$. It turns out that the compatibility is ruled by the values of the Dixmier angle between the orthogonal complement ${\mathcal S}^\perp$ of ${\mathcal S}$ and the closure of $A{\mathcal S}$. We show that every redundant decomposition ${\mathcal H}={\mathcal S}+{\mathcal M}^\perp$ (where redundant means that ${\mathcal S}\cap{\mathcal M}^\perp$ is not trivial) occurs in the presence of a certain compatibility. We also show applications of these results to some signal processing problems (consistent reconstruction) and to abstract splines problems which come from approximation theory.

Citation

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G. Corach. A. Maestripieri. "Redundant decompositions, angles between subspaces and oblique projections." Publ. Mat. 54 (2) 461 - 484, 2010.

Information

Published: 2010
First available in Project Euclid: 28 June 2010

zbMATH: 1204.46015
MathSciNet: MR2675933

Subjects:
Primary: 41A65 , 46C05 , 47A62 , 94A12

Keywords: abstract splines , angles between subspaces , compatibility , Oblique projections

Rights: Copyright © 2010 Universitat Autònoma de Barcelona, Departament de Matemàtiques

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Vol.54 • No. 2 • 2010
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