Glasnik Matematicki, Vol. 50, No. 2 (2015), 415-427.

ON EXCESSES OF FRAMES

Damir Bakić and Tomislav Berić

Department of Mathematics, University of Zagreb, 10 000 Zagreb, Croatia
e-mail: dbakic@math.hr
e-mail: tberic@math.hr


Abstract.   We show that any two frames in a separable Hilbert space that are dual to each other have the same excess. Some new relations for the analysis resp. synthesis operators of dual frames are also derived. Then we prove that pseudo-dual frames and, in particular, approximately dual frames have the same excess. We also discuss various results on frames in which excesses of frames play an important role.

2010 Mathematics Subject Classification.   42C15.

Key words and phrases.   Frame, Parseval frame, excess.


Full text (PDF) (free access)

DOI: 10.3336/gm.50.2.10


References:

  1. J. Antezana, G. Corach, M. Ruiz and D. Stojanoff, Oblique projections and frames, Proc. Amer. Math. Soc. 134 (2006), 1031-1037.
    MathSciNet     CrossRef

  2. D. Bakić and T. Berić, Finite extensions of Bessel sequences, arXiv:1308.5709v1.
    MathSciNet     CrossRef

  3. R. Balan, P.G. Casazza, D. Edidin and G. Kutyniok, A new identity for Parseval frames, Proc. Amer. Math. Soc. 135 (2007), 1007-1015.
    MathSciNet     CrossRef

  4. R. Balan, P.G. Casazza and Z. Landau, Redundancy for localized frames, Israel J. Math. 185 (2011), 445-476.
    MathSciNet     CrossRef

  5. R. Balan and Z. Landau, Measure functions for frames, J. Funct. Anal. 252 (2007), 630-676.
    MathSciNet     CrossRef

  6. R. Balan, P.G. Casazza, C. Heil and Z. Landau, Deficits and excesses of frames, Adv. Comput. Math. Special Issue on Frames, 18 (2003), 93-116.
    MathSciNet     CrossRef

  7. P.G. Casazza, The art of frame theory, Taiwanese J. Math. 4 (2000), 129-201.
    MathSciNet    

  8. P.G. Cassaza and O. Christensen, Frames containing a Riesz basis and preservation of this property under perturbations, SIAM J. Math. Anal. 29 (1998), 266-278.
    MathSciNet     CrossRef

  9. P.G. Cassaza and O. Christensen, Perturbations of operators and applications to frame theory, J. Fourier Anal. Appl. 3 (5) (1997) 543-557.
    MathSciNet     CrossRef

  10. O. Christensen, An introduction to frames and Riesz bases, Birkäuser, 2003.
    MathSciNet     CrossRef

  11. O. Christensen, A Paley-Wiener theorem for frames, Proc. Amer. Math. Soc. 123 (1995), 2199-2201.
    MathSciNet     CrossRef

  12. O. Christensen and R.S. Laugesen, Approximately dual frames in Hilbert spaces and applications to Gabor frames, http://arxiv.org/abs/0811.3588v1.

  13. I. Daubechies, Ten lectures on wavelets, SIAM, Philadelphia, 1992.
    MathSciNet     CrossRef

  14. R.J. Duffin and A.C. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc. 72 (1952), 341-366.
    MathSciNet     CrossRef

  15. P. Găvruţa, On some identities and inequalities for frames in Hilbert spaces, J. Math. Anal. Appl. 321 (2006), 469-478.
    MathSciNet     CrossRef

  16. D. Han, Frame representations and Parseval duals with applications to Gabor frames, Trans. Amer. Math. Soc. 360 (2008), 3307-3326.
    MathSciNet     CrossRef

  17. C. E. Heil and D. F. Walnut, Continuous and discrete wavelet transforms, SIAM Rev. 31 (1989), 628-666.
    MathSciNet     CrossRef

  18. J. Holub, Pre-frame operators, Besselian frames and near-Riesz bases in Hilbert spaces, Proc. Amer. Math. Soc. 122 (1994), 779-785.
    MathSciNet     CrossRef

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