A new characterization of the generalized inverse using projections on level sets

Babierra, Ariel L.; Reyes, Noli N.

doi:10.1016/j.jat.2018.07.002

Cited by 2 publications

(1 citation statement)

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“…Let H be a real Hilbert space with scalar product • | • and associated norm • , let x 0 ∈ H, let U and V be closed vector subspaces of H with projection operators proj U and proj V , respectively, and let p ∈ V . The basic best approximation problem minimize x − x 0 subject to x ∈ U and proj V x = p (1.1) covers a wide range of scenarios in areas such as harmonic analysis, signal processing, interpolation theory, and optics [3,22,32,35,38,40,43,52,59]. In this setting, a function of interest x ∈ H is known to lie in the subspace U and its projection p onto the subspace V is known.…”

Section: Introductionmentioning

confidence: 99%

Reconstruction of Functions from Prescribed Proximal Points

Combettes¹,

Woodstock²

2021

Preprint

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Under investigation is the problem of finding the best approximation of a function in a Hilbert space subject to convex constraints and prescribed nonlinear transformations. We show that in many instances these prescriptions can be represented using firmly nonexpansive operators, even when the original observation process is discontinuous. The proposed framework thus captures a large body of classical and contemporary best approximation problems arising in areas such as harmonic analysis, statistics, interpolation theory, and signal processing. The resulting problem is recast in terms of a common fixed point problem and solved with a new block-iterative algorithm that features approximate projections onto the individual sets as well as an extrapolated relaxation scheme that exploits the possible presence of affine constraints. A numerical application to signal recovery is demonstrated.

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