# Article

Full entry | PDF   (0.4 MB)
Keywords:
frame; Schauder frames
Summary:
Schauder frames were introduced by Han and Larson [9] and further studied by Casazza, Dilworth, Odell, Schlumprecht and Zsak [2]. In this paper, we have introduced approximative Schauder frames as a generalization of Schauder frames and a characterization for approximative Schauder frames in Banach spaces in terms of sequence of non-zero endomorphism of finite rank has been given. Further, weak* and weak approximative Schauder frames in Banach spaces have been defined. Finally, it has been proved that $E$ has a weak approximative Schauder frame if and only if $E^*$ has a weak* approximative Schauder frame.
References:
[1] Casazza, P.G.: The art of frame theory. Taiwanese J. Math. 4 (2) (2000), 129–201. MR 1757401 | Zbl 0966.42022
[2] Casazza, P.G., Dilworth, S.J., Odell, E., Schlumprecht, Th., Zsak, A.: Cofficient quantization for frames in Banach spaces. J. Math.Anal. Appl. 348 (2008), 66–86. DOI 10.1016/j.jmaa.2008.06.055 | MR 2449328
[3] Casazza, P.G., Han, D., Larson, D.R.: Frames for Banach spaces. Contemp. Math. 247 (1999), 149–182. DOI 10.1090/conm/247/03801 | MR 1738089 | Zbl 0947.46010
[4] Christensen, O.: Frames and bases (An introductory course). Birkhäuser, Boston, 2008. MR 2428338 | Zbl 1152.42001
[5] Daubechies, I., Grossmann, A., Meyer, Y.: Painless non-orthogonal expansions. J. Math. Phys. 27 (1986), 1271–1283. DOI 10.1063/1.527388 | MR 0836025
[6] Duffin, R.J., Schaeffer, A.C.: A class of non-harmonic Fourier series. Trans. Amer. Math. Soc. 72 (1952), 341–366. DOI 10.1090/S0002-9947-1952-0047179-6 | MR 0047179
[7] Feichtinger, H.G., Grochenig, K.: A unified approach to atomic decompostion via integrable group representations. Lecture Notes in Math., vol. 1302, Springer, Berlin, 1988, pp. 429–457. MR 0942257
[8] Gabor, D.: Theory of communications. J. Inst. Elec. Engg. 93 (1946), 429–457.
[9] Han, D., Larson, D.R.: Frames, bases and group representations. Mem. Amer. Math. Soc. 147 697) (2000), 1–91. MR 1686653 | Zbl 0971.42023
[10] Kaushik, S.K., Sharma, S.K., Poumai, K.T.: On Schauder frames in conjugate Banach spaces. J. Math. 2013 (2013), 4, Article ID 318659. MR 3096803 | Zbl 1277.46009
[11] Liu, R.: On shrinking and boundedly complete Schauder frames of Banach spaces. J. Math. Anal. Appl. 365 (1), 385–398. DOI 10.1016/j.jmaa.2009.11.001 | MR 2585111 | Zbl 1195.46012
[12] Liu, R., Zheng, B.: A characterization of Schauder frames which are near Schauder bases. J. Fourier Anal. Appl. 16 (2010), 791–803. DOI 10.1007/s00041-010-9126-5 | MR 2673710 | Zbl 1210.46012
[13] Singer, I.: Bases in Banach spaces II. Springer, New York, 1981. MR 0610799 | Zbl 0467.46020
[14] Vashisht, L.K.: On $\phi$ Schauder frames. TWMS J. Appl. Eng. Math. 2 (1) (2012), 116–120. MR 3068866 | Zbl 1274.42080

Partner of