DOI: http://dx.doi.org/10.18273/revint.v34n1-2016005

Construction and coupling of frames in
Hilbert spaces with W-metrics

GERMAN ESCOBAR^a, KEVIN ESMERAL^b, OSMIN FERRER^b*

^a Universidad Surcolombiana, Programa de Matemática Aplicada, Neiva, Colombia.
^b Universidad de Sucre, Departamento de Matemáticas, Sincelejo, Colombia.

Dedicated to the memory of our dear friend
Ricardo Cedeño

Abstract. A definition of frames unitarily equivalent in Hilbert spaces with W-metric is stated, and a characterization is given in terms of their respective analysis operators. From a Hilbert space with a frame we construct a Hilbert space with W-metric and a frame unitarily equivalent to the given one. Finally, we prove that the coupling of two frames is a frame.

Keywords: Krein spaces, W-spaces, W-metrics, construction of frames, coupling of frames, similar frames.
MSC2010: 42C15, 47B50, 46C20.

Construcción y acoplamiento de marcos en
espacios de Hilbert con W-métricas

Resumen. Se definen marcos unitariamente equivalentes en espacios de Hilbert con W-métricas, y se da una caracterización de ellos comparando sus respectivos operadores de análisis. A partir de un espacio de Hilbert con un marco se construye un espacio de Hilbert con W-métrica y un marco unitariamente equivalente al dado. Finalmente, se muestra que el acoplamiento de dos marcos es un marco.

Palabras clave: Espacios de Krein, W-espacios, W-métricas, construcción de marcos, acoplamiento de marcos, marcos similares.

Texto Completo disponible en PDF

References

[1] Acosta-Humánez P., Esmeral K. and Ferrer O., "Frames of subspaces in Hilbert spaces with W-metris", An. Stiint. Univ. Ovidius Constanta Ser. Mat. 23 (2015), No. 2, 5-22.         [ Links ]

[2] Ali S.T., Antoine J.P. and Gazeau J.P., Coherent states, wavelets and their generalizations, Springer-Verlag, New York, 2000.         [ Links ]

[3] Ali S.T., Antoine J.P. and Gazeau J.P., "Continuous frames in Hilbert space", Ann. Physis 222 (1993), No. 1, 1-37.         [ Links ]

[4] Ali S.T., Antoine J.P. and Gazeau J.P., "Square integrability of group representations on homogeneus spaces. I. Reproduing triples and frames", Ann. Inst. H. Poinaré Phys. Théor. 55 (1991), No. 4 , 829-855.         [ Links ]

[5] Azizov T.Y. and Iokhvidov I.S., "Linear operators in Hilbert spaces with G-metri", Russian Math. Surveys 26 (1971), No. 4 45-97.         [ Links ]

[6] Azizov T.Y. and Iokhvidov I.S., Linear operator in spaces with an indefinite metric, John Wiley & Sons Ltd., Chihester, 1989.         [ Links ]

[7] Bognár J., Indefinite inner product spaces, Springer-Verlag, New York-Heidelberg, 1974.         [ Links ]

[8] Casazza P.G., "The art of frame theory", Taiwanese J. Math. 4 (2000), No. 2, 129-201.         [ Links ]

[9] Casazza P.G. and Leon M.T., "Existene and onstrution of finite frames with a given frame operator", Int. J. Pure Appl. Math. 63 (2010), No. 2, 149-157.         [ Links ]

[10] Christensen O., An introdution to frames and Riesz bases, Applied and Numerical Harmonic Analysis, Birkhäuser Boston In., Boston, MA, 2003.         [ Links ]

[11] Daubechies I., "The wavelet transform, time-frequeny localization and signal analysis", IEEE Trans. Inform. Theory 36 (1990), No. 5, 961-1005.         [ Links ]

[12] Daubehies I., Grossmann A. and Meyer Y., "Painless nonorthogonal expansions", J. Math. Phys. 27 (1986), No. 5, 1271-1283.         [ Links ]

[13] Duffin R.J. and Shaeffer A.C., "A class of nonharmoni Fourier series", Trans. Amer. Math. Soc. 72 (1952), 341-366.         [ Links ]

[14] Esmeral K., Ferrer O. and Wagner E., "Frames in Krein spaces arising from a non-regular W-metric", Banah J. Math. Anal. 9 (2015), No. 1 , 1-16.         [ Links ]

[15] Gavruta P., "On the duality of fusion frames", J. Math. Anal. Appl. 333 (2007), No. 2, 871-879.         [ Links ]

[16] Giribet J.I., Maestripieri A., Martínez Pería F. and Massey P.G., "On a family of frames for Krein spaces", arXiv:1112.1632v1.         [ Links ]

[17] Giribet J.I., Maestripieri A., Martínez Pería F. and Massey P.G., "On frames for Krein spaces", J. Math. Anal. Appl. 393 (2012), No. 1, 122-137.         [ Links ]

[18] Han D., Kornelson K., Larson D.R. and Weber E., Frames for undergraduates, American Mathematical Society, Providence, RI, 2007.         [ Links ]

[19] Han D. and Larson D.R., "Frames, bases and group representations", Mem. Amer. Math. Soc. 147 (2000) No. 697, 94 pp.         [ Links ]

[20] Peng I. and Waldron S., "Signed frames and Hadamard products of Gram matrices", Linear Algebra Appl. 347 (2002), 131-157.         [ Links ]

[21] Rahimi A., Najati, A. and Dehghan Y.N., "Continuous frames in Hilbert spaces", Methods Funct. Anal. Topology 12 (2006), No. 2, 170-182.         [ Links ]

^*E-mail: osmin.ferrer@unisucre.edu.co.
Received: 18 August 2015, Accepted: 08 March 2016.
To cite this article: G. Escobar, K. Esmeral, O. Ferrer, Construction and coupling of frames in Hilbert spaces with W-metrics, Rev. Integr. Temas Mat. 34 (2016), No. 1, 81-93.