• Amiri, Zahra (Department of Pure Mathematics Ferdowsi University of Mashhad) ;
  • Kamyabi-Gol, Rajab Ali (Department of Pure Mathematics Ferdowsi University of Mashhad Center of Excellence in Analysis on Algebraic Structures (CEAAS))
  • Received : 2015.11.14
  • Published : 2017.01.01


In this paper, we study some equivalence relations between continuous frames in a Hilbert space ${\mathcal{H}}$. In particular, we seek two necessary and sufficient conditions under which two continuous frames are near. Moreover, we investigate a distance between continuous frames in order to acquire the closest and nearest tight continuous frame to a given continuous frame. Finally, we implement these results for shearlet and wavelet frames in two examples.


Supported by : University of Mashhad-Graduate Studies


  1. S. T. Ali, J. P. Antoine, and J. P. Gazeau, Continuous frames in Hilbert spaces, Ann. Physics 222 (1993), no. 1, 1-37.
  2. S. T. Ali, J. P. Antoine, and J. P. Gazeau, Coherent States Wavelets and Their Generalizations, New York, Springer-Verlag, 2000.
  3. A. A. Arefijamaal, R. A. Kamyabi-Gol, R. Raisi Tousi, and N. Tavallaei, A new approach to continuous riesz bases, J. Sci. I. R. Iran 24 (2013), 63-69.
  4. R. Balan, Equivalence relations and distances between Hilbert frames, Amer. Math. Soc. 127 (1999), no. 8, 2353-2366.
  5. O. Christensen, An Introduction to frames and Riesz Bases, Birkhauser, 2002.
  6. R. J. Duffin and A. C. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc. 72 (1952), 341-366.
  7. G. B. Folland, A Course in Abstract Harmonic Analysis, Boca Katon, CRC Press, 1995.
  8. J. P. Gabardo and D. Han, Frames associated with measurable space, Adv. Comput. Math. 18 (2003), no. 2-4, 127-147.
  9. P. Grohs, Continuous shearlet tight frames, J. Fourier Anal. Appl. 17 (2011), no. 3, 506-518.
  10. G. Kaiser, A Friendly Guide to Wavelets, Birkhauser, Boston, 1994.
  11. R. A. Kamyabi-Gol and V. Atayi, Abstract shearlet transform, Bull. Belg. Math. Soc. Simon Stevin 22 (2015), 669-681.
  12. P. Kittpoom, Irregular shearlet frames, Ph.D, thesis, 2009.
  13. G. Kutyniok and D. Labate, Construction of regular and irregular shearlet frames, J. Wavelet Theory Appl. 1 (2007), 1-12.
  14. G. Kutyniok and D. Labate, Shearlets Multiscale Analysis for Multivariate Data, Birkhauser-Springer, 2012.
  15. G. Kutyniok, D. Labate, W. Q. Lim, and G. Weiss, Sparse multidimensional representation using shearlets, In Wavelets XI (2005), 254-262.