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A Geometrical Approach to Indefinite Least Squares Problems

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Abstract

Given Hilbert spaces ℋ and \(\mathcal {K}\) , a (bounded) closed range operator \(C:\mathcal {H}\rightarrow \mathcal {K}\) and a vector \(y\in \mathcal {K}\) , consider the following indefinite least squares problem: find u∈ℋ such that 〈B(Cu−y),Cu−y〉=min x∈ℋ〈B(Cx−y),Cx−y〉, where \(B:\mathcal {K}\rightarrow \mathcal {K}\) is a bounded selfadjoint operator.

This work is devoted to give necessary and sufficient conditions for the existence of solutions of this abstract problem. Although the indefinite least squares problem has been thoroughly studied in finite dimensional spaces, the geometrical approach presented in this manuscript is quite different from the analytical techniques used before. As an application we provide some new sufficient conditions for the existence of solutions of an ℋ∞ estimation problem.

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Authors and Affiliations

  1. Departamento de Matemática, FI-UBA, Buenos Aires, Argentina

    Juan Ignacio Giribet & Alejandra Maestripieri

  2. IAM-CONICET, Saavedra 15, 3rd floor, 1083, Buenos Aires, Argentina

    Juan Ignacio Giribet, Alejandra Maestripieri & Francisco Martínez Pería

  3. Departamento de Matemática, FCE-UNLP, La Plata, Argentina

    Francisco Martínez Pería

Authors
  1. Juan Ignacio Giribet
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  2. Alejandra Maestripieri
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  3. Francisco Martínez Pería
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Corresponding author

Correspondence to Francisco Martínez Pería.

Additional information

Partially supported by PIP 5272 (CONICET), UBACyT I023, ANPCyT 1728 PICT06.

Partially supported by PIP 5272 (CONICET), UBACyT I023.

Partially supported by PIP 5272 (CONICET), UNLP 11 X472.

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Giribet, J.I., Maestripieri, A. & Martínez Pería, F. A Geometrical Approach to Indefinite Least Squares Problems. Acta Appl Math 111, 65–81 (2010). https://doi.org/10.1007/s10440-009-9532-3

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  • DOI: https://doi.org/10.1007/s10440-009-9532-3

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