Navegar

Colección

Datos Estadísticas

Artículo

Estamos trabajando para incorporar este artículo al repositorio
Consulte el artículo en la página del editor
Consulte la política de Acceso Abierto del editor

Abstract:

Given Hilbert spaces ℋ and K, a (bounded) closed range operator C: ℋ → K and a vector y ∈ 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:K → 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. © 2009 Springer Science+Business Media B.V.

Registro:

Documento: Artículo
Título:A geometrical approach to indefinite least squares problems
Autor:Giribet, J.I.; Maestripieri, A.; Martínez Pería, F.
Filiación:Departamento de Matemática, FI-UBA, Buenos Aires, Argentina
IAM-CONICET, Saavedra 15, 3rd floor, Buenos Aires 1083, Argentina
Departamento de Matemática, FCE-UNLP, La Plata, Argentina
Palabras clave:Least squares; Oblique projections; Selfadjoint operators; Weighted generalized inverses; Analytical techniques; Estimation problem; Existence of Solutions; Finite dimensional space; Indefinite least squares; Least Square; Oblique projections; Self adjoint operator; Sufficient conditions; Weighted generalized inverse; Hilbert spaces; Vector spaces
Año:2010
Volumen:111
Número:1
Página de inicio:65
Página de fin:81
DOI: http://dx.doi.org/10.1007/s10440-009-9532-3
Título revista:Acta Applicandae Mathematicae
Título revista abreviado:Acta Appl Math
ISSN:01678019
CODEN:AAMAD
Registro:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_01678019_v111_n1_p65_Giribet

Referencias:

  • Ben-Israel, A., Greville, T.N.E., (2003) Generalized Inverse: Theory and Applications, , Springer Berlin 1026.15004
  • Bognár, J., (1974) Indefinite Inner Product Spaces, , Springer Berlin 0286.46028
  • Chandrasekaran, S., Gu, M., Sayed, A.H., A stable and efficient algorithm for the indefinite least squares problem (1998) SIAM J. Matrix Anal. Appl., 20 (2), pp. 354-362. , 0920.65024 10.1137/S0895479896302229 1646860
  • Chipman, J.S., On least squares with insufficient observations (1964) J. Am. Stat. Assoc., 59, pp. 1078-1111. , 0144.42401 10.2307/2282625 175220
  • Chipman, J.S., Specification problems in regression analysis (1968) Proc. Sympos. Theory and Application of Generalized Inverses of Matrices (Lubbock, Texas, 1968), pp. 114-176. , Texas Tech. Press Lubbock
  • Corach, G., Maestripieri, A., Weighted generalized inverses, oblique projections, and least-squares problems (2005) Numerical Functional Analysis and Optimization, 26 (6), pp. 659-673. , DOI 10.1080/01630560500323083, PII L035172744233
  • Corach, G., Maestripieri, A., Stojanoff, D., Oblique projections and Schur complements (2001) Acta Sci. Math. (Szeged), 67, pp. 337-256. , 0980.47021 1830147
  • Corach, G., Maestripieri, A., Stojanoff, D., Oblique projections and abstract splines (2002) J. Approx. Theory, 117, pp. 189-206. , 1055.47016 10.1006/jath.2002.3696 1924651
  • Corach, G., Maestripieri, A., Stojanoff, D., A classification of projectors (2004) Banach Cent. Publ., 67, pp. 145-160. , 2143922
  • Dixmier, J., Etudes sur les variétés et opérateurs de Julia, avec quelques applications (1949) Bull. Soc. Math. France, 77, pp. 11-101. , 0045.39102 32937
  • Eldén, L., Perturbation theory for the least squares problem with linear equality constraints (1980) SIAM J. Numer. Anal., 17 (3), pp. 338-350. , 0469.65023 10.1137/0717028 581481
  • Golub, G.H., Van Loan, C.F., An analysis of the total least squares problem (1980) SIAM J. Numer. Anal., 17 (6), pp. 883-893. , 0468.65011 10.1137/0717073 595451
  • Hassi, S., Nordström, K., On projections in a space with an indefinite metric (1994) Linear Algebra Appl., 208-209, pp. 401-417. , 10.1016/0024-3795(94)90452-9
  • Hassibi, B., Sayed, A.H., Kailath, T., ℋ∞ Optimality criteria for LMS and backpropagation (1994) Adv. Neural Inf. Process. Syst., 6, pp. 351-359
  • Hassibi, B., Sayed, A.H., Kailath, T., Linear estimation in Krein spaces - Part I: Theory (1996) IEEE Transactions on Automatic Control, 41 (1), pp. 18-33. , PII S0018928696003868
  • Hassibi, B., Sayed, A.H., Kailath, T., Linear estimation in Krein spaces-Part II: Application (1996) IEEE Trans. Autom. Control, 41 (1), pp. 33-49
  • Hassibi, B., Sayed, A.H., Kailath, T., (1999) Indefinite-Quadratic Estimation and Control. A Unified Approach to ℋ2 and ℋ∞ Theories, , Studies in Applied and Numerical Mathematics SIAM Philadelphia
  • Iokhvidov, I.S., Azizov, T.Ya., (1989) Linear Operators in Spaces with An Indefinite Metric, , Wiley New York
  • Maestripieri, A., Martínez Pería, F., Decomposition of selfadjoint projections in Krein spaces (2006) Acta Sci. Math., 72, pp. 611-638. , 1122.47036
  • Mary, X., Moore-Penrose inverse in Krein spaces (2008) Integral. Equ. Oper. Theory, 60 (3), pp. 419-433. , 1167.47004 10.1007/s00020-008-1562-0 2392835
  • Morley, T.D., A Gauss-Markov theorem for infinite-dimensional regression models with possibly singular covariance (1979) SIAM J. Appl. Math., 37 (2), pp. 257-260. , 0438.62055 10.1137/0137016 543944
  • Nagpal, K.M., Khargonekar, P.P., Filtering and smoothing in an ℋ∞ setting (1991) IEEE Trans. Autom. Control, 36, pp. 152-166. , 0758.93074 10.1109/9.67291 1093089
  • Rao, C., Mitra, S.K., Theory and application of constrained inverse of matrices (1973) SIAM J. Appl. Math., 24, pp. 473-488. , 0237.15001 10.1137/0124050 316466
  • Sayed, A.H., Hassibi, B., Kailath, T., Fundamental Inertia Conditions for the Minimization of Quadratic Forms in Indefinite Metric Spaces (1996) Oper. Theory: Adv. Appl., , Birkhauser Cambridge
  • Sheng, X., Chen, G., The generalized weighted Moore-Penrose inverse (2007) Journal of Applied Mathematics and Computing, 25 (1-2), pp. 407-413. , DOI 10.1007/BF02832365

Citas:

---------- APA ----------
Giribet, J.I., Maestripieri, A. & Martínez Pería, F. (2010) . A geometrical approach to indefinite least squares problems. Acta Applicandae Mathematicae, 111(1), 65-81.
http://dx.doi.org/10.1007/s10440-009-9532-3
---------- CHICAGO ----------
Giribet, J.I., Maestripieri, A., Martínez Pería, F. "A geometrical approach to indefinite least squares problems" . Acta Applicandae Mathematicae 111, no. 1 (2010) : 65-81.
http://dx.doi.org/10.1007/s10440-009-9532-3
---------- MLA ----------
Giribet, J.I., Maestripieri, A., Martínez Pería, F. "A geometrical approach to indefinite least squares problems" . Acta Applicandae Mathematicae, vol. 111, no. 1, 2010, pp. 65-81.
http://dx.doi.org/10.1007/s10440-009-9532-3
---------- VANCOUVER ----------
Giribet, J.I., Maestripieri, A., Martínez Pería, F. A geometrical approach to indefinite least squares problems. Acta Appl Math. 2010;111(1):65-81.
http://dx.doi.org/10.1007/s10440-009-9532-3