2-K-Frames in 2-Hilbert Spaces

Article Sidebar

PDF
DOI: https://doi.org/10.28924/2291-8639-24-2026-101
Published: Apr 7, 2026
Cite as:
Samira Tatouti, Nordine Bounader, 2-K-Frames in 2-Hilbert Spaces, Int. J. Anal. Appl., 24 (2026), 101.

Main Article Content

Samira Tatouti, Nordine Bounader

Abstract

The aim of this article, we introduce a 2-K-frames in the 2-Hilbert space H, where K is a θ-bounded linear operator on H for a fixed element θ in H, and explore some properties of them. Also, we will characterize the 2-Kframes by the 2 pre-K-frame operators, the 2-K-frame operators, likewise to what is seen in the case of Hilbert spaces. In the rest of the article, we will set a definition of θ-atomic systems for K, and give some results concerning this notion. Finally we will find out a representation of each element in a closed range R(K).

Article Details

References

  1. A.A. Arefijamaal, G. Sadeghi, Frames in 2-Inner Product Spaces, Iran. J. Math. Sci. Inform. 8 (2013), 123-130.
  2. H. Bolcskei, F. Hlawatsch, H.G. Feichtinger, Frame-Theoretic Analysis of Oversampled Filter Banks, IEEE Trans. Signal Process. 46 (1998), 3256–3268. https://doi.org/10.1109/78.735301.
  3. Y.J. Cho, P.C.S. Lin, S.S. Kim, A. Misiak, Theory of 2-Inner Product Spaces, Nova Science Publishes, Inc., New York. 2001.
  4. O. Christensen, Frames and Bases: An Introductory Course, Birkhäuser, Boston, 2008.
  5. I. Daubechies, A. Grossmann, Y. Meyer, Painless Nonorthogonal Expansions, J. Math. Phys. 27 (1986), 1271–1283. https://doi.org/10.1063/1.527388.
  6. R.J. Duffin, A.C. Schaeffer, A Class of Nonharmonic Fourier Series, Trans. Amer. Math. Soc. 72 (1952), 341–366. https://doi.org/10.1090/S0002-9947-1952-0047179-6.
  7. Y.C. Eldar, Sampling with Arbitrary Sampling and Reconstruction Spaces and Oblique Dual Frame Vectors, J. Fourier Anal. Appl. 9 (2003), 77–96. https://doi.org/10.1007/s00041-003-0004-2.
  8. Y.C. Eldar, T. Werther, General Framework for Consistent Sampling in Hilbert Spaces, Int. J. Wavelets Multiresolution Inf. Process. 03 (2005), 497–509. https://doi.org/10.1142/S0219691305000981.
  9. P.J.S.G. Ferreira, Mathematics for Multimedia Signal Processing II: Discrete Finite Frames and Signal Reconstruction, in: J.S. Byrnes, (ed.) Signal Processing for Multimedia, pp. 35–54. IOS Press, Amsterdam, (1999).
  10. L. Găvruţa, Frames for Operators, Appl. Comput. Harmon. Anal. 32 (2012), 139–144. https://doi.org/10.1016/j.acha.2011.07.006.
  11. P.K. Harikrishnan, P. Riyas, K.T. Ravindran, Riesz Theorems in 2-Inner Product Spaces, Novi Sad J. Math. 41 (2011), 57-61.
  12. H. Mazaheri, R. Kazemi, Some Results on 2-Inner Product Spaces, Novi Sad J. Math. 37 (2007), 35-40.
  13. G.U. Reddy, Tensor Product of 2-Frames in 2-Hilbert Spaces, Adv. Pure Math. 06 (2016), 517–522. https://doi.org/10.4236/apm.2016.68039.
  14. T. Strohmer, R.W. Heath, Grassmannian Frames with Applications to Coding and Communication, Appl. Comput. Harmon. Anal. 14 (2003), 257–275. https://doi.org/10.1016/S1063-5203(03)00023-X.
  15. N.F.D. Ward, J.R. Partington, A Construction of Rational Wavelets and Frames in Hardy–Sobolev Spaces With Applications to System Modeling, SIAM J. Control Optim. 36 (1998), 654–679. https://doi.org/10.1137/S0363012996297339.
  16. X. Xiao, Y. Zhu, L. Găvruţa, Some Properties of K-Frames in Hilbert Spaces, Results Math. 63 (2013), 1243–1255. https://doi.org/10.1007/s00025-012-0266-6.