×

Abstract splines in Krein spaces. (English) Zbl 1195.47028

Let \({\mathcal K}\) be a Krein space and \({\mathcal E}\) and \({\mathcal H}\) be Hilbert spaces. Let \(T:{\mathcal H}\to{\mathcal K}\) and \(V:{\mathcal H}\to{\mathcal E}\) be bounded surjective operators, and let \(z_0\in{\mathcal E}\) be a fixed vector. The authors present generalizations to Krein spaces of the abstract interpolation, smoothing and mixed problems proposed by Atteia in Hilbert spaces. For the indefinite abstract interpolation problem, they characterize those \(x_0\in{\mathcal H}\) for which \(Vx_0=z_0\) and
\[ [Tx_0,Tx_0]_{\mathcal K}=\min\{[Tx,Tx]_{\mathcal K}:Vx=z_0\}. \]
For the indefinite abstract smoothing problem, given \(\rho>0\), they look for the minimizers of the function \(F_\rho:{\mathcal H}\to \mathbb R\) defined by
\[ F_\rho(x)=[Tx,Tx]_{\mathcal K}+\rho\|Vx-z_0\|^2_{\mathcal E}. \]
For the indefinite abstract mixed problem, they consider bounded surjective operators \(V_1:{\mathcal H}\to{\mathcal E}_1\) and \(V_2:{\mathcal H}\to{\mathcal E}_2\) between Hilbert spaces, and then, for given \(\rho>0\) and a fixed vector \((z_1,z_2)\in {\mathcal E}_1\times{\mathcal E}_2\), they look for those vectors \(x_0\in V_1^{-1}(\{z_1\})\) which minimize the function
\[ G_\rho(x)=[Tx,Tx]_{\mathcal K}+\rho\|V_2x-z_2\|^2_{{\mathcal E}_2},\quad x\in V_1^{-1}(\{z_1\}). \]
The exposition is clear and sufficient details are included for relatively easy reading. Moreover, several examples and intermediate results and lemmas are also presented.

MSC:

47B50 Linear operators on spaces with an indefinite metric
47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.)
46C20 Spaces with indefinite inner product (Kreĭn spaces, Pontryagin spaces, etc.)
41A15 Spline approximation
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] Schoenberg, I. J., Contributions to the problem of approximation of equidistant data by analytic functions. Parts A, B, Q. J. Math., 4, 45-99 (1946), 112-141
[2] Hou, H. S.; Andrews, H. C., Cubic splines for image interpolation and digital filtering, IEEE Trans. ASSP, 26, 508-517 (1978) · Zbl 0413.94011
[3] Torachi, K.; Yang, S.; Kamada, M.; Mori, R., Two dimensional spline interpolation for image reconstruction, Patt. Recognition, 21, 275-284 (1998)
[4] Unser, M.; Aldroubi, A.; Eden, M., Fast B-spline transform for continuous image reconstruction and interpolation, IEEE Trans. Patt. Anal. Machine Intell., 13, 277-285 (1991)
[5] Giribet, J. I.; España, M.; Miranda, C., Synthetic data for validation of navigation systems, Acta Astronaut., 60, 2, 88-95 (2007)
[6] Bartels, R. H.; Beatty, J. C.; Barsky, B. A., An Introduction to Splines for Use in Computer Graphics & Geometric Modeling (1987), Morgan Kaufmann Publishers Inc.: Morgan Kaufmann Publishers Inc. San Francisco · Zbl 0682.65003
[7] Meek, D. S.; Walton, D. J., A note on planar minimax arc splines, Comput. & Graphics, 16, 431-433 (1992) · Zbl 1050.65017
[8] Meek, D. S.; Walton, D. J., Approximating smooth planar curves by arc splines, J. Comput. Appl. Math., 59, 221-231 (1995) · Zbl 0836.65010
[11] Atteia, M., Géneralization de la définition et des propietés des “splines fonctions”, C. R. Sci. Paris, 260, 3550-3553 (1965) · Zbl 0163.37703
[12] Anselone, P. M.; Laurent, P. J., A general method for the construction of interpolating or smoothing spline-functions, Numer. Math., 12, 66-82 (1968) · Zbl 0197.13501
[13] de Boor, C., Convergence of abstract splines, J. Approx. Theory, 31, 80-89 (1981) · Zbl 0477.41012
[14] Laurent, P. J., Approximation et optimisation (1972), Hermann: Hermann Paris · Zbl 0238.90058
[15] Sard, A., Optimal approximation, J. Funct. Anal.. J. Funct. Anal., J. Funct. Anal., 2, 368-369 (1968), (addendum) · Zbl 0159.43801
[16] Rozhenko, A. I., Mixed spline approximation, Bull. Novosibirsk Comput. Center Ser.: Numer. Anal., 5, 67-86 (1994) · Zbl 0906.41010
[17] Atteia, M., Hilbertian Kernels and Spline Functions (1992), North-Holland Publishing Co.: North-Holland Publishing Co. Amsterdam · Zbl 0767.41015
[18] Bezhaev, A. Yu.; Vasilenko, V. A., Variational Theory of Splines (2001), Kluwer Academic/Plenum Publishers: Kluwer Academic/Plenum Publishers New York · Zbl 0979.41007
[19] Champion, R.; Lenard, C. T.; Mills, T. M., An introduction to abstract splines, Math. Sci., 21, 8-26 (1996) · Zbl 0851.41033
[20] Hassi, S.; Nordström, K., On projections in a space with an indefinite metric, Linear Algebra Appl., 208/209, 401-417 (1994) · Zbl 0803.46022
[22] Rozhenko, A. I.; Vasilenko, V. A., Variational approach in abstract splines: achievements and open problems, East J. Approx., 1, 277-308 (1995) · Zbl 0852.41008
[23] Bognár, J., Indefinite Inner Product Spaces (1974), Springer-Verlag · Zbl 0277.47024
[24] Iokhvidov, I. S.; Azizov, T. Ya., Linear Operators in Spaces with an Indefinite Metric (1989), John Wiley & Sons · Zbl 0714.47028
[25] Ando, T., Linear Operators on Krein Spaces (1979), Hokkaido University: Hokkaido University Sapporo, Japan · Zbl 0429.47016
[26] Dritschel, M. A.; Rovnyak, J., Operators on indefinite inner product spaces, (Lancaster, Peter, Fields Inst. Monogr., vol. 3 (1996), Amer. Math. Soc.), 141-232 · Zbl 0883.47019
[27] Rovnyak, J., Methods on Krein space operator theory, (Interpolation Theory, Systems Theory and Related Topics. Interpolation Theory, Systems Theory and Related Topics, Tel Aviv/Rehovot, 1999. Interpolation Theory, Systems Theory and Related Topics. Interpolation Theory, Systems Theory and Related Topics, Tel Aviv/Rehovot, 1999, Oper. Theory Adv. Appl., vol. 134 (2002)), 31-66 · Zbl 1062.47002
[28] Deutsch, F., The angle between subspaces of a Hilbert space, (Approximation Theory, Wavelets and Applications. Approximation Theory, Wavelets and Applications, Maratea, 1994. Approximation Theory, Wavelets and Applications. Approximation Theory, Wavelets and Applications, Maratea, 1994, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 454 (1995), Kluwer Acad. Publ.: Kluwer Acad. Publ. Dordrecht), 107-130 · Zbl 0848.46010
[29] Bouldin, R., The product of operators with closed range, Tohoku Math. J., 25, 359-363 (1973) · Zbl 0269.47002
[30] Izumino, S., The product of operators with closed range and an extension of the reverse order law, Tohoku Math. J., 34, 43-52 (1982) · Zbl 0481.47001
[31] Kato, T., Perturbation Theory for Linear Operators (1966), Springer-Verlag: Springer-Verlag New York · Zbl 0148.12601
[33] Casazza, P.; Christensen, O., Frames containing a Riesz basis and preservation of this property under perturbations, SIAM J. Math. Anal., 29, 1, 266-278 (1998) · Zbl 0922.42024
[34] Walnut, D. F., An Introduction to Wavelets Analysis (2004), Springer-Verlag
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.