Results in Mathematics

, 74:191 | Cite as

Characterization of Alternate Duals of Continuous Frames and Representation Frames

  • Ali Akbar ArefijamaalEmail author
  • Atefe Razghandi


We give a new characterization of alternate dual of continuous frames and then apply these results to representation frames which are generated by a unitary representation on a locally compact group. We also investigate under which conditions alternate duals can be reduced to the canonical dual. Finally, we introduce representation frames with a unique representation form dual and obtain more results on alternate (not necessary representation form) duals.


Dual frames continuous frames unitary representation representation frames 

Mathematics Subject Classification

Primary 42C15 Secondary 20C99 



  1. 1.
    Ali, S.T., Antoine, J.P., Gazeau, J.P.: Coherent States, Wavelets and Their Generalizations. Springer, New York (2000)CrossRefGoogle Scholar
  2. 2.
    Arefijamaal, A.A., Arabyani Neyshaburi, F., Shamsabadi, M.: On the duality of frames and fusion frames. Hacet. J. Math. Stat. 47(1), 47–56 (2018)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Arefijamaal, A.A., Zekaee, E.: Signal processing by alternate dual Gabor frames. Appl. Comput. Harmon. Anal. 35, 535–540 (2013)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Balazs, P., Stoeva, D.T.: Representation of the inverse of a frame multiplier. J. Math. Anal. Appl. 422(2), 981–994 (2015)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Casazza, P., Kovačević, J.: Equal-norm tight frames with erasures. Adv. Comput. Math. 18, 387–430 (2003)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Christensen, O.: Frames and Bases: An Introductory Course. Birkhäuser, Boston (2008)CrossRefGoogle Scholar
  7. 7.
    Daubechies, I.: The wavelet transform, time–frequency localization and signal analysis. IEEE Trans. Inform. Theory 36, 961–1005 (1990)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Daubechies, I., Grossmann, A., Meyer, Y.: Painless nonorthogonal expansions. J. Math. Phys. 27, 1271–1283 (1986)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Eldar, Y., Bölcskei, H.: Geometrically uniform frames. IEEE Trans. Inform. Theory 49, 993–1006 (2003)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Folland, G.B.: A Course in Abstract Harmonic Analysis. CRC Press, Boca Raton (1995)zbMATHGoogle Scholar
  11. 11.
    Gabardo, J.P., Han, D.: Frame representations for group- like unitary operator systems. J. Oper. Theory 49, 223–244 (2003)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Gabardo, J.P., Han, D.: The uniqueness of the dual of Weyl–Heisenberg subspace frames. Appl. Comput. Harmon. Anal. 17, 226–240 (2004)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Gabardo, J.P., Han, D.: Frame associated with measurable spaces. Adv. Comput. Math. 18(3), 127–147 (2003)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Grossmann, A., Morlet, J., Paul, T.: Transforms associated to square integrable group representations. I. General results. J. Math. Phys. 26, 2473–2479 (1985)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Han, D.: Approximations for Gabor and wavelet frames. Trans. Am. Math. Soc. 355, 3329–3342 (2003)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Han, D.: Frame representations and parseval duals with applications to Gabor frames. Trans. Am. Math. Soc. 360, 3307–3326 (2008)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Han, D., Larson, D.: Frames bases and group representations. Mem. Am. Math. Soc. 147(697), (2000)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Han, D., Larson, D., Liu, B., Liu, R.: Operator-valued measures, Dilations, and the Theory of Frames. Mem. Am. Math. Soc. 229(1075), (2014)Google Scholar
  19. 19.
    Heil, C.E., Walnut, D.F.: Continuous and discrete wavelet transforms. SIAM Rev. 31, 628–666 (1989)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Kaiser, G.: A Friendly Guide to Wavelets. Birkhauser, Boston (1994)zbMATHGoogle Scholar
  21. 21.
    Li, Z., Han, D.: Frames vector multipliers for finite group representations. Linear Algebra Appl. 519, 191–207 (2017)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Lopez, J., Han, D.: Optimal dual frames for erasures. Linear Algebra Appl. 432, 471–482 (2010)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Mallat, S.: A Wavelet Tour of Signal Processing, 2nd edn. Academic Press, Cambridge (1999)zbMATHGoogle Scholar
  24. 24.
    Rahimi, A., Najati, A., Dehghan, Y.N.: Continuous frames in Hilbert spaces. Methods Funct. Anal. Topol. 12(2), 170–182 (2006)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematics and Computer SciencesHakim Sabzevari UniversitySabzevarIran

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