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A CHARACTERIZATION OF WAVE PACKET FRAMES FOR L2 Rd

Year 2018, Volume: 8 Issue: 2, 353 - 361, 01.12.2018

Abstract

In this paper we present necessary and sucient conditions with explicit frame bounds for a nite sum of wave packet frames to be a frame for L2 Rd . Further, we illustrate our results with some examples and applications.

References

  • Casazza, P. G., Kutyniok, G., (2012), Finite frames: Theory and Applications, Birkh¨auser.
  • Cerone, P., Dragomir, S.S., (2011), Mathematical Inequalities, CRC Press, New York.
  • Christensen, O., Linear combinations of frames and frame packets, Z. Anal. Anwend., 20 (4), pp. 815. Christensen, O., (2002), An introduction to frames and Riesz bases, Birkh¨auser, Boston.
  • Christensen, O., Rahimi, A., (2008), Frame properties of wave packet systems in L2(R), Adv. Comput. Math., 29, pp. 101–111.
  • Cordoba, A., Fefferman, C., (1978), Wave packets and Fourier integral operators, Comm. Partial Differential Equations, 3 (11), pp. 979–1005.
  • Czaja, W., Kutyniok, G., Speegle, D., (2006), The Geometry of sets of prameters of wave packets, Appl. Comput. Harmon. Anal., 20, pp. 108–125.
  • Heil, C., Walnut, D. (1989), Continuous and discrete wavelet transforms, SIAM Rev., 31 (4), pp. –666.
  • Heil, C., (2011), A basis theory primer, Expanded edition. Applied and Numerical Harmonic Analysis.
  • Birkh¨auser/Springer, New York. Hern´andez, E., Labate, D., Weiss, G., (2002), A unified characterization of reproducing systems generated by a finite family II, J. Geom. Anal., 12 (4), pp. 615–662.
  • Hern´andez, E., Labate, D., Weiss, G., Wilson, E., (2004), Oversampling, quasi-affine frames and wave packets, Appl. Comput. Harmon. Anal., 16, pp. 111–147.
  • Kumar, R., Sah, A. K., (2016), Stability of multivariate wave packet frames for L2(Rn), Boll. Unione Mat. Ital., DOI 10.1007/s40574-016-0106-9.
  • Kumar, R., Sah, A. K., (2016), Matrix Transform of Irregular Weyl-Heisenberg Wave Packet Frames for L2(R), TWMS J. App. Eng. Math., Accepted.
  • Kumar, R., Sah, A. K., (2017), Perturbation of Irregular Weyl-Heisenberg Wave Packet Frames in
  • L2(R), Osaka J. Math., Preprint. Labate, D., Weiss, G., Wilson, E., (2004), An approach to the study of wave packet systems, Contemp. Math., 345, pp. 215–235.
  • Lacey, M., Thiele, C., (1997), Lpestimates on the bilinear Hilbert transform for 2 < p < ∞, Ann. Math., 146, pp. 69–724.
  • Lacey, M., Thiele, C., (1999), On Calder´on’s conjecture, Ann. Math., 149, pp. 475–496.
  • Sah, A.K., (2016) Linear combination of wave packet frame for L2(Rd), Wavelets and Linear Algebra, (2), pp. 19–32.
  • Sah, A.K., Vashisht, L.K., (2014), Hilbert transform of irregular wave packet system for L2(R)
  • Poincare J. Anal. Appl., 1, pp. 9–17. Sah, A.K., Vashisht, L.K., (2015), Irregular Weyl-Heisenberg wave packet frames in L2(R), Bull. Sci. Math. 139, pp. 61–74.
  • Ashok Kumar Sah for the photography and short autobiography, see TWMS J. App. Eng. Math. V.7, N.2, 2017.
Year 2018, Volume: 8 Issue: 2, 353 - 361, 01.12.2018

Abstract

References

  • Casazza, P. G., Kutyniok, G., (2012), Finite frames: Theory and Applications, Birkh¨auser.
  • Cerone, P., Dragomir, S.S., (2011), Mathematical Inequalities, CRC Press, New York.
  • Christensen, O., Linear combinations of frames and frame packets, Z. Anal. Anwend., 20 (4), pp. 815. Christensen, O., (2002), An introduction to frames and Riesz bases, Birkh¨auser, Boston.
  • Christensen, O., Rahimi, A., (2008), Frame properties of wave packet systems in L2(R), Adv. Comput. Math., 29, pp. 101–111.
  • Cordoba, A., Fefferman, C., (1978), Wave packets and Fourier integral operators, Comm. Partial Differential Equations, 3 (11), pp. 979–1005.
  • Czaja, W., Kutyniok, G., Speegle, D., (2006), The Geometry of sets of prameters of wave packets, Appl. Comput. Harmon. Anal., 20, pp. 108–125.
  • Heil, C., Walnut, D. (1989), Continuous and discrete wavelet transforms, SIAM Rev., 31 (4), pp. –666.
  • Heil, C., (2011), A basis theory primer, Expanded edition. Applied and Numerical Harmonic Analysis.
  • Birkh¨auser/Springer, New York. Hern´andez, E., Labate, D., Weiss, G., (2002), A unified characterization of reproducing systems generated by a finite family II, J. Geom. Anal., 12 (4), pp. 615–662.
  • Hern´andez, E., Labate, D., Weiss, G., Wilson, E., (2004), Oversampling, quasi-affine frames and wave packets, Appl. Comput. Harmon. Anal., 16, pp. 111–147.
  • Kumar, R., Sah, A. K., (2016), Stability of multivariate wave packet frames for L2(Rn), Boll. Unione Mat. Ital., DOI 10.1007/s40574-016-0106-9.
  • Kumar, R., Sah, A. K., (2016), Matrix Transform of Irregular Weyl-Heisenberg Wave Packet Frames for L2(R), TWMS J. App. Eng. Math., Accepted.
  • Kumar, R., Sah, A. K., (2017), Perturbation of Irregular Weyl-Heisenberg Wave Packet Frames in
  • L2(R), Osaka J. Math., Preprint. Labate, D., Weiss, G., Wilson, E., (2004), An approach to the study of wave packet systems, Contemp. Math., 345, pp. 215–235.
  • Lacey, M., Thiele, C., (1997), Lpestimates on the bilinear Hilbert transform for 2 < p < ∞, Ann. Math., 146, pp. 69–724.
  • Lacey, M., Thiele, C., (1999), On Calder´on’s conjecture, Ann. Math., 149, pp. 475–496.
  • Sah, A.K., (2016) Linear combination of wave packet frame for L2(Rd), Wavelets and Linear Algebra, (2), pp. 19–32.
  • Sah, A.K., Vashisht, L.K., (2014), Hilbert transform of irregular wave packet system for L2(R)
  • Poincare J. Anal. Appl., 1, pp. 9–17. Sah, A.K., Vashisht, L.K., (2015), Irregular Weyl-Heisenberg wave packet frames in L2(R), Bull. Sci. Math. 139, pp. 61–74.
  • Ashok Kumar Sah for the photography and short autobiography, see TWMS J. App. Eng. Math. V.7, N.2, 2017.
There are 20 citations in total.

Details

Primary Language English
Journal Section Research Article
Authors

Ashok K. Sah This is me

Publication Date December 1, 2018
Published in Issue Year 2018 Volume: 8 Issue: 2

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