Abstract:
We present generalizations to Krein spaces of the abstract interpolation and smoothing problems proposed by Atteia in Hilbert spaces: given a Krein space K and Hilbert spaces H and E (bounded) surjective operators T:H→K and VH→E, ρ>0 and a fixed z0∈E, we study the existence of solutions of the problems argmin{[Tx,Tx]K: Vx=z0} and argmin{[Tx,Tx]K+ρ{norm of matrix}Vx-z0{norm of matrix}E2x∈H}. © 2010 Elsevier Inc.
Registro:
Documento: |
Artículo
|
Título: | Abstract splines in Krein spaces |
Autor: | Giribet, J.I.; Maestripieri, A.; Pería, F.M. |
Filiación: | Departamento de Matemática, FI-UBA, Paseo Colón 850, (1063) Buenos Aires, Argentina Departamento de Matemática, FCE-UNLP, CC 172, (1900) La Plata, Argentina Instituto Argentino de Matemática Alberto Calderón - CONICET, Saavedra 15, Piso 3, (1083) Buenos Aires, Argentina
|
Palabras clave: | Abstract splines; Krein spaces; Oblique projections |
Año: | 2010
|
Volumen: | 369
|
Número: | 1
|
Página de inicio: | 423
|
Página de fin: | 436
|
DOI: |
http://dx.doi.org/10.1016/j.jmaa.2010.03.016 |
Título revista: | Journal of Mathematical Analysis and Applications
|
Título revista abreviado: | J. Math. Anal. Appl.
|
ISSN: | 0022247X
|
Registro: | https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_0022247X_v369_n1_p423_Giribet |
Referencias:
- Schoenberg, I.J., Contributions to the problem of approximation of equidistant data by analytic functions. Parts A, B (1946) Q. J. Math., 4, pp. 45-99. , 112-141
- Hou, H.S., Andrews, H.C., Cubic splines for image interpolation and digital filtering (1978) IEEE Trans. ASSP, 26, pp. 508-517
- Torachi, K., Yang, S., Kamada, M., Mori, R., Two dimensional spline interpolation for image reconstruction (1998) Patt. Recognition, 21, pp. 275-284
- Unser, M., Aldroubi, A., Eden, M., Fast B-spline transform for continuous image reconstruction and interpolation (1991) IEEE Trans. Patt. Anal. Machine Intell., 13, pp. 277-285
- Giribet, J.I., España, M., Miranda, C., Synthetic data for validation of navigation systems (2007) Acta Astronaut., 60 (2), pp. 88-95
- Bartels, R.H., Beatty, J.C., Barsky, B.A., (1987) An Introduction to Splines for Use in Computer Graphics & Geometric Modeling, , Morgan Kaufmann Publishers Inc., San Francisco
- Meek, D.S., Walton, D.J., A note on planar minimax arc splines (1992) Comput. & Graphics, 16, pp. 431-433
- Meek, D.S., Walton, D.J., Approximating smooth planar curves by arc splines (1995) J. Comput. Appl. Math., 59, pp. 221-231
- Canu, S., Ong, C.S., Mary, X., Smola, A., Learning with non-positive kernels, in: Proceedings of the 21st International Conference on Machine Learning (2004), pp. 639-646; Canu, S., Ong, C.S., Mary, X., Splines with non positive kernels, in: Proceedings of the 5th International ISAAC Congress, (2005), pp. 1-10; Atteia, M., Géneralization de la définition et des propietés des " splines fonctions" (1965) C. R. Sci. Paris, 260, pp. 3550-3553
- Anselone, P.M., Laurent, P.J., A general method for the construction of interpolating or smoothing spline-functions (1968) Numer. Math., 12, pp. 66-82
- de Boor, C., Convergence of abstract splines (1981) J. Approx. Theory, 31, pp. 80-89
- Laurent, P.J., (1972) Approximation et optimisation, , Hermann, Paris
- Sard, A., Optimal approximation (1968) J. Funct. Anal.. J. Funct. Anal., 2, pp. 368-369. , (addendum)
- Rozhenko, A.I., Mixed spline approximation (1994) Bull. Novosibirsk Comput. Center Ser.: Numer. Anal., 5, pp. 67-86
- Atteia, M., (1992) Hilbertian Kernels and Spline Functions, , North-Holland Publishing Co., Amsterdam
- Bezhaev, A., Vasilenko, V.A., (2001) Variational Theory of Splines, , Kluwer Academic/Plenum Publishers, New York
- Champion, R., Lenard, C.T., Mills, T.M., An introduction to abstract splines (1996) Math. Sci., 21, pp. 8-26
- Hassi, S., Nordström, K., On projections in a space with an indefinite metric (1994) Linear Algebra Appl., pp. 401-417
- Rozhenko, A.I., The conditions of unique solvability of mixed spline approximation problem, Preprint N1023, Comp. Center USSR Acad. Sci. Press, Novosibirsk, 1994 (in Russian) ; Rozhenko, A.I., Vasilenko, V.A., Variational approach in abstract splines: achievements and open problems (1995) East J. Approx., 1, pp. 277-308
- Bognár, J., (1974) Indefinite Inner Product Spaces, , Springer-Verlag
- Iokhvidov, I.S., Azizov, T., (1989) Linear Operators in Spaces with an Indefinite Metric, , John Wiley & Sons
- Ando, T., (1979) Linear Operators on Krein Spaces, , Hokkaido University, Sapporo, Japan
- Dritschel, M.A., Rovnyak, J., Operators on indefinite inner product spaces (1996) Fields Inst. Monogr., vol. 3, pp. 141-232. , Amer. Math. Soc. P. Lancaster (Ed.)
- Rovnyak, J., Methods on Krein space operator theory (2002) Oper. Theory Adv. Appl., vol. 134, pp. 31-66. , Interpolation Theory, Systems Theory and Related Topics
- Deutsch, F., The angle between subspaces of a Hilbert space (1995) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 454, pp. 107-130. , Kluwer Acad. Publ., Dordrecht, Approximation Theory, Wavelets and Applications
- Bouldin, R., The product of operators with closed range (1973) Tohoku Math. J., 25, pp. 359-363
- Izumino, S., The product of operators with closed range and an extension of the reverse order law (1982) Tohoku Math. J., 34, pp. 43-52
- Kato, T., (1966) Perturbation Theory for Linear Operators, , Springer-Verlag, New York
- Patel, H., Solving the indefinite least squares problem, PhD thesis, Univ. of Manchester, (2002); Casazza, P., Christensen, O., Frames containing a Riesz basis and preservation of this property under perturbations (1998) SIAM J. Math. Anal., 29 (1), pp. 266-278
- Walnut, D.F., (2004) An Introduction to Wavelets Analysis, , Springer-Verlag
Citas:
---------- APA ----------
Giribet, J.I., Maestripieri, A. & Pería, F.M.
(2010)
. Abstract splines in Krein spaces. Journal of Mathematical Analysis and Applications, 369(1), 423-436.
http://dx.doi.org/10.1016/j.jmaa.2010.03.016---------- CHICAGO ----------
Giribet, J.I., Maestripieri, A., Pería, F.M.
"Abstract splines in Krein spaces"
. Journal of Mathematical Analysis and Applications 369, no. 1
(2010) : 423-436.
http://dx.doi.org/10.1016/j.jmaa.2010.03.016---------- MLA ----------
Giribet, J.I., Maestripieri, A., Pería, F.M.
"Abstract splines in Krein spaces"
. Journal of Mathematical Analysis and Applications, vol. 369, no. 1, 2010, pp. 423-436.
http://dx.doi.org/10.1016/j.jmaa.2010.03.016---------- VANCOUVER ----------
Giribet, J.I., Maestripieri, A., Pería, F.M. Abstract splines in Krein spaces. J. Math. Anal. Appl. 2010;369(1):423-436.
http://dx.doi.org/10.1016/j.jmaa.2010.03.016