Document Type : Research Paper


Department of mathematics, University of Maragheh, Maragheh, Iran.


Motivating the perturbations of frames in Hilbert and Banach spaces, in this paper we introduce the invariance of Fr'echet frames under perturbation. Also we show that for any Fr'echet spaces, there is a Fr'echet frame and any element in these spaces  has a series expansion.


[1]  A. Aldroubi, Q. Sun and W. Tang, p-frames and shift invariant subspaces of Lp, J. Fourier Anal. Appl. 7 no. 1, (2001) 1-22.
[2] P. Balazs, D. Bayer, A. Rahimi, Multipliers for continuous frames in Hilbert spaces, Journal of Physics A: Mathematical and Theoretical, 45 (2012).
[3] P. G. Casazza and O. Christensen, Perturbation of operators and application to frame theory, J. Fourier Anal. Appl. 3(5) (1997) 543{557.
[4] P. G. Casazza, O. Christensen and D. T. Stoeva, Frame expansions in separable Banach spaces, J. Math. Anal. Appl. 307 (2005) 710723.
[5] P. G. Casazza, D. Han and D. Larson, Frames for Banach spaces, Contemp. Math. 247 (1999) 149182.
[6] O. Christensen, An Introduction to Frame and Riesz Bases, Birkhauser 2002.
[7] O. Christensen, Frame perturbations, Proc. Amer. Math. Soc. 123 (1995) 12171220.
[8] O. Christensen, A Paley-Wiener theorem for frames, Proc. Amer. Math. Soc. 123 (1995) 21992201.
[9] R. J. Dun and A. C. Schae er, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc. 72 (1952) 341-366.
[10] K. Grochenig, Describing functions: atomic decompositions versus frames, Monatsh. Math. 112 (1) (1991) 142.
[11] P. K. Jain, Sh. K. Kaushik and L. K. Vashisht, On Stability of Banach Frames, Bull. Korean Math. Soc. 44 (2007) 73-81.
[12] T. Kato, Perturbation Theory for Linear Operators, Springer, New York, 1976.
[13] R. Meise, D. Vogt, Introduction to Functional Analysis, Clarendon Press, Oxford, 1997.
[14] A. Najati and A. Rahimi, Generalized frames in Hilbert spaces, Bull. Iran. Math. Soc. Vol. 35 No. 1 (2009) 97-109.
[15] S. Pilipovic, D.T. Stoeva and N. Teofanov, Frames for Frechet spaces, Bull. Cl. Sci. Math. Nat., 32 (2007) 6984.
[16] S. Pilipovic and D.T. Stoeva, Series expansions in Frechet spaces and their duals; construction of Frechet frames, Journal of Approximation Theory 163 (2011) 17291747.
[17] A. Rahimi, Frames and Their Generalizations in Hilbert and Banach Spaces , Lambert Academic Publishing, 2011.
[18] A. Rahimi and A. H. Fereydooni, Controlled G-Frames And Their G-Multipliers In Hilbert Spaces, Anal. Stii. Univ. OVIDIUS CON. Seria Math, ( accepted paper).
[19] A. Rahimi, A. Najati and Y. N. Dehghan, Continuous frames in Hilbert spaces, Meth. Func. Anal. Top. Vol. 12, No. 2 (2006) 170-182.
[20] D. T. Stoeva, Perturbation of frames in Banach spaces, Asian- European Journal of Mathematics 5, N0.1 (2012), 1250011 (15 pages), DOI: 10.1142/S1793557112500118.