Document Type : Research Paper

Authors

1 Department of Non-harmonic analysis, Institute of Mathematics and Mechanics of NAS of Azerbaijan, Baku, Azerbaijan.

2 Department of Functional analysis, Institute of Mathematics and Mechanics of NAS of Azerbaijan, Baku, Azerbaijan.

Abstract

The generalization of p-frame in Banach spaces is considered in this paper. The concepts of an $\tilde{X}$-frame and a system conjugate to $\tilde{X}$-frame were introduced. Analogues of the results on the existence of conjugate system were obtained. The stability of $\tilde{X}$-frame having a conjugate system is studied.

Keywords

[1] R. J. Duffin and A. C. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc., 72  (1952)  341-–366.
[2] R. Young, An introduction to nonharmonic Fourier series,  New York, 1980.
[3] O. Christensen, An Introduction to Frames and Riesz Bases,  Appl. Numer. Harmon. Anal., Birkhauser, Boston, MA, 2002.
[4] P. G. Casazza and  O. Christensen, Perturbation operators and applications to frame theory,  J. Fourier Anal. Appl.,  3(5) (1997) 543--557.
[5] H. G. Feichtinger  and  K. H. Gröchenig, Banach spaces related to integrable group representations and their atomic decomposition,  J. Funct. Anal., 86 (1989)  307--340.
[6] R. Balan, Stability theorems for Fourier frames and wavelet Riesz bases,  J. Fourier Anal. And Appl., 3 (1997)   499--504.
[7] P. G. Casazza, The art of frame theory,  Taiwanese J. Math., 4(2) (2000)  129--201.
[8] K. Gröchenig, Describing functions: atomic decomposition versus frames, Monatsh. Math., 112(1) (1991) 1--41.
[9] O. Christensen  and C. Heil, Perturbations of Frames and Atomic Decompositions,  Math. Nachr., 185 (1997)  33--47.
[10] P. G. Casazza, D.  Han  and D. R. Larson, Frames for Banach space,  Contemp. Math., 247 (1999)  149--182.
[11] P. K. Jain, S. K. Kaushik and  L. K. Vashisht, On perturbations of Banach frames,  Int. J. Wavelet multiresolut.Inf. Process, 4(3) (2006)   559--565.
[12] A. Aldroubi, Q. Sun and W. Tang,  p-frames and shift invariant subspaces of  Lp,  J. Fourier Anal. Appl., 7 (2001)  1--21.
[13] O. Christensen and D.  Stoeva,  p-frames in separable Banach spaces, Adv. Comp. Math., 18 (2003) 117--126.
[14] W. Sun, G-frames and G-Riesz bases,  J. Math. Anal. Appl.,  322 (2006)  473--452.
[15] M.R. Abdollahpour,  M.H. Faroughi  and  A. Rahimi, PG-frames in Banach spaces,  Methods of functional Analysis and Topology, 13(3) (2007) 201--210.
 
[16]  B.T. Bilalov and  F.A.Guliyeva, On The Frame Properties of Degenerate System of Sines, Journal of Function Spaces and Applications, 2012 (2012) Article ID 184186, 12 pages, doi:10.1155/2012/184186.
 
[17] S.R. Sadigova  and  Z.V. Mamedova, Frames from Cosines with the Degenerate Coefficients,  American Journal of Applied Mathematics and Statistics, 1(3) (2013) 36--40.
 
[18] Gar J. Garnett, Bounded Analytic Functions, Moscow, "Mir", 1984, 469 p.