Series expansions in Fréchet spaces and their duals, construction of Fréchet frames. (English) Zbl 1231.42030
The authors investigate Banach frames and Fréchet spaces \(X_F\) with respect to Fréchet sequence spaces \(\theta_F\). Results on series expansions in Fréchet spaces and their duals are obtained.
Reviewer: Julian Musielak (Poznań)
MSC:
| 42C15 | General harmonic expansions, frames |
| 42C40 | Nontrigonometric harmonic analysis involving wavelets and other special systems |
References:
| [1] | Albiac, F.; Kalton, N., Topics in Banach Space Theory (2006), Springer · Zbl 1094.46002 |
| [2] | Aldroubi, A.; Sun, Q.; Tang, W., \(p\)-frames and shift invariant subspaces of \(L^p\), J. Fourier Anal. Appl., 7, 1-21 (2001) · Zbl 0983.46027 |
| [3] | Aldroubi, A.; Sun, Q.; Tang, W., Connection between \(p\)-frames and \(p\)-Riesz bases in locally finite SIS of \(L^p(R)\), Proc. SPIE, 4119, 668-674 (2000) |
| [4] | Borup, L.; Gribonval, R.; Nielsen, M., Tight wavelet frames in Lebesgue and Sobolev spaces, J. Funct. Spaces Appl., 2, 227-252 (2004) · Zbl 1058.42025 |
| [5] | Casazza, P. G.; Han, D.; Larson, D. R., Frames for Banach spaces, Contemp. Math., 247, 149-182 (1999) · Zbl 0947.46010 |
| [6] | Casazza, P. G.; Christensen, O.; Stoeva, D. T., Frame expansions in separable Banach spaces, J. Math. Anal. Appl., 307, 710-723 (2005) · Zbl 1091.46007 |
| [7] | Christensen, O.; Heil, C., Perturbations of Banach frames and atomic decompositions, Math. Nachr., 185, 33-47 (1997) · Zbl 0868.42013 |
| [8] | Cordero, E.; Gröchenig, K., Localization of frames II, Appl. Comput. Harmonic Anal., 17, 29-47 (2004) · Zbl 1061.42016 |
| [9] | Feichtinger, H. G.; Gröchenig, K., Banach spaces related to integrable group representations and their atomic decompositions I, J. Funct. Anal., 86, 307-340 (1989) · Zbl 0691.46011 |
| [10] | Feichtinger, H. G.; Gröchenig, K., Banach spaces related to integrable group representations and their atomic decompositions II, Monatsh. Math., 108, 129-148 (1989) · Zbl 0713.43004 |
| [11] | Fornasier, M.; Gröchenig, K., Intrinsic localization of frames, Constr. Approx., 22, 395-415 (2005) · Zbl 1130.41304 |
| [12] | Gröchenig, K., Localized frames are finite unions of Riesz sequences, Adv. Comput. Math., 18, 149-157 (2003) · Zbl 1019.42019 |
| [13] | Gröchenig, K., Describing functions: frames versus atomic decompositions, Monatsh. Math., 112, 1-41 (1991) · Zbl 0736.42022 |
| [14] | Gröchenig, K., Localization of frames, Banach frames, and the invertibility of the frame operator, J. Fourier Anal. Appl., 10, 105-132 (2004) · Zbl 1055.42018 |
| [15] | C. Heil, A basis theory primer, Manuscript, 1997. Available online at http://www.math.gatech.edu/ heil/papers/bases.pdf; C. Heil, A basis theory primer, Manuscript, 1997. Available online at http://www.math.gatech.edu/ heil/papers/bases.pdf |
| [16] | Kantorovich, L. V.; Akilov, G. P., Functional Analysis in Normed Spaces (1964), Pergamon Press · Zbl 0127.06104 |
| [17] | Köthe, G., Topological Vector Spaces I (1969), Springer-Verlag · Zbl 0179.17001 |
| [18] | Meise, R.; Vogt, D., Introduction to Functional Analysis (1997), Clarendon Press: Clarendon Press Oxford · Zbl 0924.46002 |
| [19] | Pilipović, S., Generalization of Zemanian spaces of generalized functions which have orthonormal series expansions, SIAM J. Math. Anal., 17, 477-484 (1986) · Zbl 0604.46042 |
| [20] | Pilipović, S.; Stoeva, D. T.; Teofanov, N., Frames for Fréchet spaces, Bull. Cl. Sci. Math. Nat. Sci. Math., 32, 69-84 (2007) · Zbl 1249.42023 |
| [21] | Singer, I., Bases in Banach spaces I (1970), Springer-Verlag: Springer-Verlag New York · Zbl 0198.16601 |
| [22] | Stoeva, D. T., \(X_d\)-frames in Banach spaces and their duals, Int. J. Pure Appl. Math., 52, 1-14 (2009) · Zbl 1173.42322 |
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