Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

DISTANCE BETWEEN CONTINUOUS FRAMES IN HILBERT SPACE

  • 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
Download PDF
⟨ Previous Next ⟩

Abstract

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.

Keywords

References

  1. S. T. Ali, J. P. Antoine, and J. P. Gazeau, Continuous frames in Hilbert spaces, Ann. Physics 222 (1993), no. 1, 1-37. https://doi.org/10.1006/aphy.1993.1016
  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. https://doi.org/10.1090/S0002-9939-99-04826-1
  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. https://doi.org/10.1090/S0002-9947-1952-0047179-6
  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. https://doi.org/10.1023/A:1021312429186
  9. P. Grohs, Continuous shearlet tight frames, J. Fourier Anal. Appl. 17 (2011), no. 3, 506-518. https://doi.org/10.1007/s00041-010-9149-y
  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.