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Redundant decompositions, angles between subspaces and oblique projections. (English) Zbl 1204.46015

Authors’ abstract: Let \({\mathcal H}\) be a complex Hilbert space. The authors 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 \(A{\mathcal S}\). They 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.

MSC:

46C05 Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product)
47A62 Equations involving linear operators, with operator unknowns
94A12 Signal theory (characterization, reconstruction, filtering, etc.)
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
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Full Text: DOI Euclid