Rocky Mountain Journal of Mathematics

Some constructions of $K$-frames and their duals

Fahimeh Arabyani Neyshaburi and Ali Akbar Arefijamaal

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

$K$-frames, as a new generalization of frames, have important applications, especially in sampling theory, to help us to reconstruct elements from a range of a bounded linear operator $K$ in a separable Hilbert space. In this paper, we focus on the reconstruction formulae to characterize all $K$-duals of a given $K$-frame. Also, we give several approaches for constructing $K$-frames.

Article information

Source
Rocky Mountain J. Math., Volume 47, Number 6 (2017), 1749-1764.

Dates
First available in Project Euclid: 21 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1511254951

Digital Object Identifier
doi:10.1216/RMJ-2017-47-6-1749

Mathematical Reviews number (MathSciNet)
MR3725243

Zentralblatt MATH identifier
1384.42024

Subjects
Primary: 42C15: General harmonic expansions, frames
Secondary: 41A58: Series expansions (e.g. Taylor, Lidstone series, but not Fourier series)

Keywords
$K$-frames $K$-duals $K$-minimal frames

Citation

Neyshaburi, Fahimeh Arabyani; Arefijamaal, Ali Akbar. Some constructions of $K$-frames and their duals. Rocky Mountain J. Math. 47 (2017), no. 6, 1749--1764. doi:10.1216/RMJ-2017-47-6-1749. https://projecteuclid.org/euclid.rmjm/1511254951


Export citation

References

  • A. Aldroubi, Portraits of frames, Proc. Amer. Math. Soc. 123 (1995), 1661–1668.
  • A. Arefijamaal and E. Zekaee, Signal processing by alternate dual Gabor frames, Appl. Comp. Harmon. Anal. 35 (2013), 535–540.
  • B.G. Bodmannand and V.I. Paulsen, Frames, graphs and erasures, Linear Alg. Appl. 404 (2005), 118–146.
  • H. Bolcskel, F. Hlawatsch and H.G. Feichtinger, Frame-theoretic analysis of oversampled filter banks, IEEE Trans. Signal Proc. 46 (1998), 3256–3268.
  • E.J. Candes and D.L. Donoho, New tight frames of curvelets and optimal representations of objects with piecewise $C^2$ singularities, Comm. Pure Appl. Anal. 56 (2004), 216–266.
  • P.G. Casazza, The art of frame theory, Taiwanese J. Math. 4 (2000), 129–202.
  • O. Christensen, Frames and bases: An introductory course, Birkhäuser, Boston, 2008.
  • ––––, Extensions of Bessel sequences to dual pairs of frames, Appl. Comp. Harmon. Anal. 34 (2013), 224–233.
  • O. Christensen and R.S. Laugesen, Approximately dual frames in Hilbert spaces and applications to Gabor frames, Samp. Theory Signal Image Process. 9 (2010), 77–89.
  • H.G. Feichtinger and K. Grochenig, Irregular sampling theorems and series expansion of band-limited functions, Math. Anal. Appl. 167 (1992), 530–556.
  • H.G. Feichtinger and T. Werther, Atomic systems for subspaces, Proc. SampTA, L. Zayed, ed., Orlando, FL, 2001.
  • L. Găvruţa, Frames for operators, Appl. Comp. Harm. Anal. 32 (2012), 139–144.
  • M. Pawlak and U. Stadtmuller, Recovering band-limited signals under noise, IEEE Trans. Inf. Theory 42 (1994), 1425–1438.
  • T. Werther, Reconstruction from irregular samples with improved locality, Masters thesis, University of Vienna, Vienna, 1999.
  • X.C. Xiao, Y.C. Zhu and L. Gavruta, Some properties of $K$-frames in Hilbert spaces, Results Math. 63 (2013), 1243–1255.
  • X.C. Xiao, Y.C. Zhu, Z.B. Shu and M.L. Ding, G-frames with bounded linear operators, Rocky Mountain Math. 45 (2015), 675–693.