Document Type : Research Paper

Author

Baku State University, Institute of Mathematics and Mechanics of the NAS of Azerbaijan.

Abstract

Some generalizations of Besselian, Hilbertian systems and frames in nonseparable Banach spaces with respect to some nonseparable Banach space $K$ of systems of scalars are considered in this work. The concepts of  uncountable $K$-Bessel, $K$-Hilbert systems, $K$-frames and  $K^{*} $-Riesz bases in nonseparable Banach spaces are introduced. Criteria of uncountable $K$-Besselianness, $K$-Hilbertianness for systems, $K$-frames and unconditional $K^{*} $-Riesz basicity are found, and the relationship between them is studied. Unlike before, these new facts about Besselian and Hilbertian systems in Hilbert and Banach spaces are proved without using a conjugate system and, in some cases, a completeness of a system. Examples of $K$-Besselian systems which are not minimal are given. It is proved that every $K$-Hilbertian systems is minimal. The case where $K$ is an space of systems of coefficients of uncountable unconditional basis of some space is also considered.

Keywords

[1] A. Aldroubi, Q. Sun and W. Tang, $p$-frames and shift invariant subspaces of $L_p $, J. Fourier Anal. Appl., 7(2001), pp. 1-21.
[2] M.R. Abdollahpour, M.H. Faroughi and A. Rahimi, $PG$-frames in Banach spaces, Methods Funct. Anal. Topology, 13 (3) (2007), pp. 201-210.
[3] N.K. Bari, Biorthogonal systems and bases in Hilbert space, Mosc. Gos. Univ. Uc. Zap. 148, Matematika 4 (1951), pp. 69-107. (in Russian)
[4] B.T. Bilalov and F.A. Guliyeva, On The Frame Properties of Degenerate System of Sines, J. Funct. Spaces Appl., 2012 (2012), 12 pages.
[5] B.T. Bilalov and F.A. Guliyeva, Neotherian perturbation of frames, Pensee Int. J., 75:12 (2013), pp. 425-431.
[6] B.T. Bilalov and Sh.M. Hashimov, On Decomposition In Banach Spaces, Proc. Inst. Math. Mech. Natl. Acad. Sci. Azerb., 40 (2) (2014), pp. 97-106.
[7] B.T. Bilalov and F.A. Guliyeva, t-Frames and their Noetherian Perturbation, Complex Anal. Oper. Theory, 8 (7) (2014), pp. 1405-1418.
[8] B.T. Bilalov and Z.G. Guseinov, K -Bessel and K -Hilbert systems and K-bases, Dokl. Math., 80 (3) (2009), pp. 826-828.
[9] B.T. Bilalov, M.I. Ismailov and Z.V. Mamedova, Uncountable Frames in Non-Separable Hilbert Spaces and their Characterization, Azerb. J. Math., 8 (1) (2018), pp. 151-178.
[10] Z.A. Canturija, On some properties of biorthogonal systems in Banach space and their applications in spectral theory, Soobshch. Akad. Nauk Gruz. SSR, 2:34 (1964), pp. 271-276.
[11] P.G. Casazza and O. Christensen, Perturbation of operators and applications to frame theory, J. Math. Anal. Appl., 307 (2) (2005), pp. 710-723.
[12] P.G. Casazza and O. Christensen, Approximation of the inverse frame operator and applications to Gabor frames, J. Approx. Theory 103(2)(2000), pp. 338-356.
[13] P.G. Casazza, O. Christensen and D.T. Stoeva, Frame expansions in separable Banach spaces, J. Math. Anal. Appl., 307 (2) (2005), pp. 710-723.
[14] P.G. Casazza, D. Han and D.R. Larson, Frames for Banach space, Contemp. Math., 247 (1999), pp. 149-182.
[15] O. Christensen, An Introduction to Frames and Riesz Bases, Birkhauser Boston, 2002.
[16] O. Christensen and C. Heil, Perturbations of Frames and Atomic Decompositions, Math. Nachr., 185 (1997), pp. 33-47.
[17] O. Christensen and D.T. Stoeva, $p$-frames in separable Banach spaces, Adv. Comput. Math., 18 (2003), pp. 117-126.
[18] I. Daubechies, Ten lectures on wavelets, SIAM, Philadelphia, 1992.
[19] R.J. Duffin and A.C. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc., 72 (1952), pp. 341-366.
[20] H.G. Feichtinger and K.H. Grochening, Banach spaces related to integrable group representations and their atomic decompositions, J. Func. Anal., 86 (2) (1989), pp. 307-340
[21] K. Gr"ochenig, Describing functions: atomic decomposition versus frames, Monatsh. Math., 112 (1) (1991), pp. 1-41. 
[22] Ch. Heil, A Basis Theory Primer, Springer, 2011.
[23] D. Han and D.R. Larson, Frames, bases and group representations, Memoirs Amer. Math. Soc., 147:697(2000), pp. 1-91.
[24] M.I. Ismailov and A. Jabrailova, On $tilde{X}$-frames and conjugate systems in Banach spaces, Sahand Commun. Math. Anal., 1 (2) (2014), pp. 19-26.
[25] M.I. Ismailov, Y.I. Nasibov, On One Generalization of Banach frame, Azerb. J. Math., 6 (2) (2016), pp. 143-159.
[26] M.I. Ismailov, F. Guliyeva and Y. Nasibov, On a generalization of the Hilbert frame generated by the bilinear mapping, J. Funct. Anal., (2016), pp.1-8.
[27] M.I. Ismailov, On Bessel and Riesz-Fisher systems with respect to Banach space of vector-valued sequences, Bull. Transilv. Univ. of Brasov, Ser. III, 12:61 (2) (2019), pp. 303-318.
[28] M.I. Ismailov, On uncountable $K$-Bessel and $K$-Hilbert systems in nonseparable Banach space, Proc. Inst. Math. Mech. Natl. Acad. Sci. Azerb., 45 (2) (2019), pp. 192-204.
[29] P.K. Jain, S.K. Kaushik, and L.K. Vashisht, On stability of Banach frames, Bull. Korean Math. Soc., 44 (1) (2007), pp.73-81.
[30] Y. Meyer, Wavelets and operators, Herman, Paris, 1990.
[31] A. Pelczunski and I. Singer, On non-equivalent bases and conditional bases in Banach spaces, Studia Math., 25 (1964), pp.5-25.
[32] A. Rahimi, Frames and Their Generalizations in Hilbert and Banach Spaces, Lap Lambert Academic Publ., 2011.
[33] W. Sun, Stability of g-frames, J. Math. Anal.Appl., 326 (2) (2007), pp. 858-868.
[34] W. Sun, G-frames and g-Riesz bases, J. Math. Anal.Appl., 322 (1)(2006), pp. 437-452.
[35] P.A. Terekhin, On Besselian systems in a Banach space, Mat. Zametki, 91:2 (2012), pp. 285-296. (in Russian)
[36] Veits B.E. Bessel and Hilbert systems in Banach spaces and stability problems, Izv. Vyssh. Uchebn. Zaved. Mat., 2:45 (1965), pp. 7-23. (in Russian)
[37] R. Young, An introduction to nonharmonic Fourier series, Academic Press, New York, 1980.