Some Results of Frames in Krein Spaces

Document Type : Research Paper

Authors

1 Vali-e-Asr University of Rafsanjan

2 Department of Mathematic, Vali-e-Asr University

10.22103/jmmrc.2021.17086.1131

Abstract

In recently years, frames in Krein spaces had been considered. The paper presents a family
of generators for a Krein space by their frames. These generators are dual frames and operator dual frames
corresponding to a given frame in a Krein space. We characterize all generalized dual frames of a primary frame. Also, approximately
dual frames in a Krein space are introduced and, we study the relation between approximately dual frames and operator duals
in a Krein space. Finally, perturbation of frames in this space is considered.

Keywords


[1] A. Aldroubi, Q. Sun, and W.S. Tang, P-frames and shift invariant subspaces of LP , J. Fourier Anal.
Appl., Vol. 7(2001), 1-21.
[2] T. Ya. Azizov and I.S. Iokhvidov, Linear Operator in Spaces with an Indefinite Metric, Wiley-
Interscience, Chichester, 1989.
[3] O. Christensen, An Introduction to Frames and Riesz Bases, Birkh¨auser, 2016.
[4] O. Christensen and R.S. Laugesen, Approximately dual frame pairs in Hilbert spaces and applications
to Gabor frames, Sampl. Theory Signal Image Process., Vol. 9(2011), 77-90.
[5] I. Daubechies, A. Grossmann, and Y. Meyer, Painless nonorthogonal expansions, J. Math. Phys., Vol.
27(1986), 1271-1383.
[6] M.A. Dehghan and M.A. Hasankhani Fard, G-dual frames in Hilbert spaces, U. P. B. Sci. Bull., Series
A, Vol. 75(2013), 129-140.
[7] R.J. Duffin and A.C. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc., Vol.
72(1952), 341-366.
[8] K. Esmeral, O. Ferrer, and E. Wagner, Frames in Krein spaces arising from a non-regular w-metric, J.
Math. Anal., Vol. 9(2015), 1-16.
[9] Ju. Ginzberg, On J-contractive operator functions, Dokl. Akad. Nauk, SSSR, Vol. 117(1957), 171-173.
[10] J.I. Giribet, A. Maestripier, F. Martnez Per´ıa, and P.G. Massey, On frames for Krein spaces, J. Math.
Anal. Appl., Vol. 393(2012), 122-137.
[11] C. Heil, Wiener amalgam space in generalized harmonic analysis and wavelet theory, Ph.D. Thesis,
University of Maryland, College Park, 1990.
[12] S. Karmakar and SK.M. Hossein, Frames on Krein spaces, Math.(2014), Arxiv:1406-6205.
[13] J. Kovacevic, P.L. Dragotti, and V. Goyal, Filter Bank Frame Expansions with Erasures, IEEE Trans.
Information Theory, Vol. 48, No. 6(2012), 1439-1450.
[14] C.P. Lewis, Frames and frame perturbations in Hilbert spaces, M.S. Thesis, University of Pittsburgh,
1996.
[15] D.G. Mixon, Sparse Signal Processing with Frame Theory, Ph.D. Thesis, Faculty of Princeton University
in Candidacy, 2012.
[16] L. Pontrjagin, Hermitian operators in spaces with indefinite metric, Izv. Akad. Nauk. SSSR, Ser. Mat.,
Vol. 8(1944), 243-280.
[17] J. Weidmann, Linear Operators in Hilbert Spaces, Springer Verlag, New York, 1980.