Certain Characterizations of Tight Gabor Frames on Local Fields



Frame, local field, Gabor frames.


Gabor systems are generated by modulations and translations of a single function. Many researchers studied Gabor frames in Hilbert spaces. The concepts Gabor frames on local fields, first introduced by Li and Jiang. They studied the existence of a Gabor frame on local fields and also established some necessary conditions and two sufficient conditions of Gabor frame for local fields. Inspired by above paper, in this paper, we study certain characterizations of tight Gabor frames on the local fields of positive characteristic.

URL http://dspace.chitkara.edu.in/jspui/bitstream/1/542/1/32010_MJIS_Abdullah.pdf
DOI 10.15415/mjis.2015.32010
  • Abdullah and Shah, F.A. Wave packet frames on local fields of positive characteristic. Appl. Math. Comput., 249 , 133-141 (2014).
  • [2] Abdullah. Tight wave packet frames for L 2 ()  and H 2 ()  . Arab J. Math. Sci., 19 (2), 151–158 (2013).
  • Casazza, P.G. The art of frame theory. Taiwanese J. Math., 4 , 129-201 (2000).
  • Christensen, O. Frames, Riesz bases, and discrete Gabor/wavelet expansions. Bull. Amer. Math. Soc. (New Series), 38 (3), 273-291 (2001).
  • Christensen, O. (2003). An Introduction to Frames and Riesz Bases. Birkhauser, Boston.
  • Duffin, R.J. and Schaeffer, A.C. A class of nonharmonic Fourier series. Trans. Amer. Math. Soc., 72 , 341-366 (1952).
  • Daubechies, I., Grossmann, A. and Meyer, Y. Painless nonorthogonal expansions. J. Math. Phys., 27 , 1271-1283 (1986).
  • Gabor, D. Theory of communications. Jour. Inst. Elec. Eng., 93 , 429-457 (1946).
  • Jiang, H.K., Li, D.F. and Jin, N. Multiresolution analysis on local fields. J. Math. Anal. Appl., 294 (2), 523-532 (2004).
  • Li, D. and Jiang, H. Basic results of Gabor frame on local fields. Chin. Ann. Math., 28 b (2), 65-176 (2007).
  • Shah, F.A. Gabor frames on a half-line. J. contemp. Math. Anal., 47 (5), 251-260, (2012).
  • Shah, F.A. and Abdullah. Necessary condition for the existence of wave packet frames. Southeast Asian Bull. of Maths, 36 , 287-292 (2012).
  • Taibleson, M.H. (1975). Fourier Analysis on Local Fields, Princeton University Press, Princeton.