On the a b c -problem in Weyl-Heisenberg frames

Xinggang He; Haixiong Li

Czechoslovak Mathematical Journal (2014)

  • Volume: 64, Issue: 2, page 447-458
  • ISSN: 0011-4642

Abstract

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Let a , b , c > 0 . We investigate the characterization problem which asks for a classification of all the triples ( a , b , c ) such that the Weyl-Heisenberg system { e 2 π i m b x ⁢ χ [ n a , n a + c ) : m , n ∈ ℤ } is a frame for L 2 ( ℝ ) . It turns out that the answer to the problem is quite complicated, see Gu and Han (2008) and Janssen (2003). Using a dilation technique, one can reduce the problem to the case where b = 1 and only let a and c vary. In this paper, we extend the Zak transform technique and use the Fourier analysis technique to study the problem for the case of a being a rational number. We prove some special cases of values for c and a that do not produce a frame, which expands earlier works.

How to cite

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He, Xinggang, and Li, Haixiong. "On the $abc$-problem in Weyl-Heisenberg frames." Czechoslovak Mathematical Journal 64.2 (2014): 447-458. <http://eudml.org/doc/262053>.

@article{He2014,
abstract = {Let $a,b,c>0$. We investigate the characterization problem which asks for a classification of all the triples $(a,b,c)$ such that the Weyl-Heisenberg system $\lbrace \{\rm e\}^\{2\pi \{\rm i\}mbx\} \* \chi _\{[na,na+c)\}\colon m,n\in \{\mathbb \{Z\}\}\rbrace $ is a frame for $L^2(\{\mathbb \{R\}\})$. It turns out that the answer to the problem is quite complicated, see Gu and Han (2008) and Janssen (2003). Using a dilation technique, one can reduce the problem to the case where $b=1$ and only let $a$ and $c$ vary. In this paper, we extend the Zak transform technique and use the Fourier analysis technique to study the problem for the case of $a$ being a rational number. We prove some special cases of values for $c$ and $a$ that do not produce a frame, which expands earlier works.},
author = {He, Xinggang, Li, Haixiong},
journal = {Czechoslovak Mathematical Journal},
keywords = {$abc$-problem; Weyl-Heisenberg frame; Zak transform; $abc$-problem; Weyl-Heisenberg frame; Zak transform},
language = {eng},
number = {2},
pages = {447-458},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the $abc$-problem in Weyl-Heisenberg frames},
url = {http://eudml.org/doc/262053},
volume = {64},
year = {2014},
}

TY - JOUR
AU - He, Xinggang
AU - Li, Haixiong
TI - On the $abc$-problem in Weyl-Heisenberg frames
JO - Czechoslovak Mathematical Journal
PY - 2014
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 64
IS - 2
SP - 447
EP - 458
AB - Let $a,b,c>0$. We investigate the characterization problem which asks for a classification of all the triples $(a,b,c)$ such that the Weyl-Heisenberg system $\lbrace {\rm e}^{2\pi {\rm i}mbx} \* \chi _{[na,na+c)}\colon m,n\in {\mathbb {Z}}\rbrace $ is a frame for $L^2({\mathbb {R}})$. It turns out that the answer to the problem is quite complicated, see Gu and Han (2008) and Janssen (2003). Using a dilation technique, one can reduce the problem to the case where $b=1$ and only let $a$ and $c$ vary. In this paper, we extend the Zak transform technique and use the Fourier analysis technique to study the problem for the case of $a$ being a rational number. We prove some special cases of values for $c$ and $a$ that do not produce a frame, which expands earlier works.
LA - eng
KW - $abc$-problem; Weyl-Heisenberg frame; Zak transform; $abc$-problem; Weyl-Heisenberg frame; Zak transform
UR - http://eudml.org/doc/262053
ER -

References

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  2. Casazza, P. G., Kalton, N. J., 10.1090/S0002-9939-02-06352-9, Proc. Am. Math. Soc. 130 (2002), 2313-2318. (2002) Zbl0991.42023MR1896414DOI10.1090/S0002-9939-02-06352-9
  3. Gröchenig, K., Stöckler, J., 10.1215/00127094-2141944, Duke Math. J. 162 (2013), 1003-1031. (2013) Zbl1277.42037MR3053565DOI10.1215/00127094-2141944
  4. Gu, Q., Han, D., 10.1016/j.acha.2007.06.005, Appl. Comput. Harmon. Anal. 24 (2008), 290-309. (2008) Zbl1242.42023MR2407006DOI10.1016/j.acha.2007.06.005
  5. Heil, C., 10.1007/s00041-006-6073-2, J. Fourier Anal. Appl. 13 (2007), 113-166. (2007) Zbl1133.42043MR2313431DOI10.1007/s00041-006-6073-2
  6. Janssen, A. J. E. M., 10.1016/0019-3577(96)85088-9, Indag. Math., New Ser. 7 (1996), 165-183. (1996) Zbl1056.42512MR1621312DOI10.1016/0019-3577(96)85088-9
  7. Janssen, A. J. E. M., Zak transforms with few zeros and the tie, Advances in Gabor Analysis H. G. Feichtinger et al. Applied and Numerical Harmonic Analysis Birkhäuser, Basel 31-70 (2003). (2003) Zbl1027.42025MR1955931
  8. Janssen, A. J. E. M., Strohmer, T., 10.1006/acha.2001.0376, Appl. Comput. Harmon. Anal. 12 (2002), 259-267. (2002) Zbl1005.42021MR1884237DOI10.1006/acha.2001.0376
  9. Lyubarskij, Y. I., Frames in the Bargmann space of entire functions, Entire and Subharmonic Functions Advances in Soviet Mathematics 11 American Mathematical Society, Providence (1992), 167-180. (1992) Zbl0770.30025MR1188007
  10. Seip, K., Density theorems for sampling and interpolation in the Bargmann-Fock space I, J. Reine Angew. Math. 429 (1992), 91-106. (1992) Zbl0745.46034MR1173117
  11. Seip, K., Wallstén, R., Density theorems for sampling and interpolation in the Bargmann-Fock space II, J. Reine Angew. Math. 429 (1992), 107-113. (1992) Zbl0745.46033MR1173118

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