Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

LOCALIZATION PROPERTY AND FRAMES II

  • HA YOUNG-HWA (Department of Mathematics Ajou University) ;
  • RYU JU-YEON (Department of Mathematics Ajou University)
  • Published : 2006.01.01
Download PDF
⟨ Previous Next ⟩

Abstract

Localization of sequences with respect to Riesz bases for Hilbert spaces are comparable with perturbation of Riesz bases or frames. Grochenig first introduced the notion of localization. We introduce more general definition of localization and show that exponentially localized sequences and polynomially localized sequences with respect to Riesz bases are Bessel sequences. Furthermore, they are frames provided some additional conditions are satisfied.

Keywords

References

  1. P. G. Casazza and O. Christensen, Approximation of the inverse frame opertor and applications to Gabor frames, J. Approx. Theory 103 (2000), 338-356 https://doi.org/10.1006/jath.1999.3350
  2. O. Christensen, An introduction to frames and Riesz basis, Birkhauser, Boston, 2003
  3. O. Christensen, A Paley-Wiener theorem for frames, Proc. Amer. Math. Soc. 123 (1995), 2199-2201
  4. O. Christensen, Frame perturbations, Proc. Amer. Math. Soc. 123 (1995), 1217-1220
  5. E. Cordero and K. Grochenig, Localization of frames II, Appl. Comput. Harmon. Anal. 17 (2004), 29-47 https://doi.org/10.1016/j.acha.2004.02.002
  6. I. Daubechies, Ten Lectures on Wavelets, SIAM Conference Series in Applied Mathematiccs, SIAM, Boston, 1992
  7. K. Grochenig, Localization of frames, Banach frames, and the invertiblility of the frame operator, J. Fourier Anal. Appl. 10 (2004), 105-132 https://doi.org/10.1007/s00041-004-8007-1
  8. K. Grochenig, Localized frames are finite unions of Riesz sequences, Adv. Comput. Math. 18 (2003), 149-157 https://doi.org/10.1023/A:1021368609918
  9. K. Grochenig, Foundations of time-frequency analysis, Birkhauser, Boston, 2001
  10. Y. H. Ha and J. Y. Ryu, Localization property and frames, Honam Math. J. 27 (2005), 233-241
  11. R. Young, An Introduction to Nonharmonic Fourier Series, Academic Press, NewYork, 1980