| Title:
|
Generalized Schauder frames (English) |
| Author:
|
Kaushik, S.K. |
| Author:
|
Sharma, Shalu |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
50 |
| Issue:
|
1 |
| Year:
|
2014 |
| Pages:
|
39-49 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
frame |
| Keyword:
|
Schauder frames |
| MSC:
|
42C15 |
| MSC:
|
42C30 |
| MSC:
|
94C15 |
| idZBL:
|
Zbl 06391564 |
| idMR:
|
MR3194767 |
| DOI:
|
10.5817/AM2014-1-39 |
| . |
| Date available:
|
2014-04-04T07:18:34Z |
| Last updated:
|
2015-03-19 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143718 |
| . |
| Reference:
|
[1] Casazza, P.G.: The art of frame theory.Taiwanese J. Math. 4 (2) (2000), 129–201. Zbl 0966.42022, MR 1757401 |
| Reference:
|
[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. MR 2449328, 10.1016/j.jmaa.2008.06.055 |
| Reference:
|
[3] Casazza, P.G., Han, D., Larson, D.R.: Frames for Banach spaces.Contemp. Math. 247 (1999), 149–182. Zbl 0947.46010, MR 1738089, 10.1090/conm/247/03801 |
| Reference:
|
[4] Christensen, O.: Frames and bases (An introductory course).Birkhäuser, Boston, 2008. Zbl 1152.42001, MR 2428338 |
| Reference:
|
[5] Daubechies, I., Grossmann, A., Meyer, Y.: Painless non-orthogonal expansions.J. Math. Phys. 27 (1986), 1271–1283. MR 0836025, 10.1063/1.527388 |
| Reference:
|
[6] Duffin, R.J., Schaeffer, A.C.: A class of non-harmonic Fourier series.Trans. Amer. Math. Soc. 72 (1952), 341–366. MR 0047179, 10.1090/S0002-9947-1952-0047179-6 |
| Reference:
|
[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 |
| Reference:
|
[8] Gabor, D.: Theory of communications.J. Inst. Elec. Engg. 93 (1946), 429–457. |
| Reference:
|
[9] Han, D., Larson, D.R.: Frames, bases and group representations.Mem. Amer. Math. Soc. 147 697) (2000), 1–91. Zbl 0971.42023, MR 1686653 |
| Reference:
|
[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. Zbl 1277.46009, MR 3096803 |
| Reference:
|
[11] Liu, R.: On shrinking and boundedly complete Schauder frames of Banach spaces.J. Math. Anal. Appl. 365 (1), 385–398. Zbl 1195.46012, MR 2585111, 10.1016/j.jmaa.2009.11.001 |
| Reference:
|
[12] Liu, R., Zheng, B.: A characterization of Schauder frames which are near Schauder bases.J. Fourier Anal. Appl. 16 (2010), 791–803. Zbl 1210.46012, MR 2673710, 10.1007/s00041-010-9126-5 |
| Reference:
|
[13] Singer, I.: Bases in Banach spaces II.Springer, New York, 1981. Zbl 0467.46020, MR 0610799 |
| Reference:
|
[14] Vashisht, L.K.: On $\phi $ Schauder frames.TWMS J. Appl. Eng. Math. 2 (1) (2012), 116–120. Zbl 1274.42080, MR 3068866 |
| . |
