Generalized Schauder frames

S.K. Kaushik; Shalu Sharma

Archivum Mathematicum (2014)

  • Volume: 050, Issue: 1, page 39-49
  • ISSN: 0044-8753

Abstract

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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.

How to cite

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Kaushik, S.K., and Sharma, Shalu. "Generalized Schauder frames." Archivum Mathematicum 050.1 (2014): 39-49. <http://eudml.org/doc/261076>.

@article{Kaushik2014,
abstract = {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.},
author = {Kaushik, S.K., Sharma, Shalu},
journal = {Archivum Mathematicum},
keywords = {frame; Schauder frames; frame; Schauder frame},
language = {eng},
number = {1},
pages = {39-49},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Generalized Schauder frames},
url = {http://eudml.org/doc/261076},
volume = {050},
year = {2014},
}

TY - JOUR
AU - Kaushik, S.K.
AU - Sharma, Shalu
TI - Generalized Schauder frames
JO - Archivum Mathematicum
PY - 2014
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 050
IS - 1
SP - 39
EP - 49
AB - 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.
LA - eng
KW - frame; Schauder frames; frame; Schauder frame
UR - http://eudml.org/doc/261076
ER -

References

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  1. Casazza, P.G., The art of frame theory, Taiwanese J. Math. 4 (2) (2000), 129–201. (2000) Zbl0966.42022MR1757401
  2. Casazza, P.G., Dilworth, S.J., Odell, E., Schlumprecht, Th., Zsak, A., 10.1016/j.jmaa.2008.06.055, J. Math.Anal. Appl. 348 (2008), 66–86. (2008) MR2449328DOI10.1016/j.jmaa.2008.06.055
  3. Casazza, P.G., Han, D., Larson, D.R., 10.1090/conm/247/03801, Contemp. Math. 247 (1999), 149–182. (1999) Zbl0947.46010MR1738089DOI10.1090/conm/247/03801
  4. Christensen, O., Frames and bases (An introductory course), Birkhäuser, Boston, 2008. (2008) Zbl1152.42001MR2428338
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  6. Duffin, R.J., Schaeffer, A.C., 10.1090/S0002-9947-1952-0047179-6, Trans. Amer. Math. Soc. 72 (1952), 341–366. (1952) MR0047179DOI10.1090/S0002-9947-1952-0047179-6
  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. (1988) MR0942257
  8. Gabor, D., Theory of communications, J. Inst. Elec. Engg. 93 (1946), 429–457. (1946) 
  9. Han, D., Larson, D.R., Frames, bases and group representations, Mem. Amer. Math. Soc. 147 697) (2000), 1–91. (2000) Zbl0971.42023MR1686653
  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. (2013) Zbl1277.46009MR3096803
  11. Liu, R., 10.1016/j.jmaa.2009.11.001, J. Math. Anal. Appl. 365 (1), 385–398. Zbl1195.46012MR2585111DOI10.1016/j.jmaa.2009.11.001
  12. Liu, R., Zheng, B., 10.1007/s00041-010-9126-5, J. Fourier Anal. Appl. 16 (2010), 791–803. (2010) Zbl1210.46012MR2673710DOI10.1007/s00041-010-9126-5
  13. Singer, I., Bases in Banach spaces II, Springer, New York, 1981. (1981) Zbl0467.46020MR0610799
  14. Vashisht, L.K., On φ Schauder frames, TWMS J. Appl. Eng. Math. 2 (1) (2012), 116–120. (2012) Zbl1274.42080MR3068866

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