Skip to main content Accessibility help
×
Home

Bessel Sequences and Its F-Scalability

  • Lei Liu (a1), Xianwei Zheng (a1), Jingwen Yan (a2) and Xiaodong Niu (a3)

Abstract

Frame theory, which contains wavelet analysis and Gabor analysis, has become a powerful tool for many applications of mathematics, engineering and quantum mechanics. The study of extension principles of Bessel sequences to frames is important in frame theory. This paper studies transformations on Bessel sequences to generate frames and Riesz bases in terms of operators and scalability. Some characterizations of operators that mapping Bessel sequences to frames and Riesz bases are given. We introduce the definitions of F-scalable and P-scalable Bessel sequences. F-scalability and P-scalability of Bessel sequences are discussed in this paper, then characterizations of scalings of F-scalable or P-scalable Bessel sequences are established. Finally, a perturbation result on F-scalable Bessel sequences is derived.

Copyright

Corresponding author

*Corresponding author. Email: wliulei@stu.edu.cn (L. Liu), 09xwzheng@stu.edu.cn (X. Zheng), jwyan@stu.edu.cn (J. Yan), xdniu@stu.edu.cn (X. Niu)

References

Hide All
[1]Bakić, D. and Berić, T., Finite extensions of Bessel sequences, ArXiv preprint arXiv:1308.5709, 2013.
[2]Balazs, P., Antoine, J. P. and Grybos, A., Weighted and controlled frames: Mutual relationship and first numerical properties, Ijwmip, 8 (2010), pp. 109132.
[3]Cahill, J. and Chen, X., A note on scalable frames, ArXiv preprint arXiv:1301.7292, 2013.
[4]Casazza, P. G., The art of frame theory, Taiwanese J. Math., 4 (2000), pp. 129202.
[5]Casazza, P. G. and Christensen, O., Perturbation of operators and applications to frame theory, J. Fourier Anal. Appl., 3 (1997), pp. 543557.
[6]Casazza, P. G. and Kutyniok, G., Finite Frames Theory and Applications, Birkhäuser, Boston, 2013.
[7]Casazza, P. G. and Leonhard, N., Classes of finite equal norm Parseval frames, Contemp. Math., 451 (2008), pp. 1131.
[8]Christensen, O., An Introduction to Frames and Riesz Bases, Birkhäuser, Boston, 2003.
[9]Christensen, O., Kim, H. O. and Kim, R. Y., Extensions of Bessel sequences to dual pairs of frames, Appl. Comput. Harmon. Anal., 34 (2013), pp. 224233.
[10]Cotfas, N. and Gazeau, J. P., Finite tight frames and some applications, J. Phys. A Math. Theor., 43 (2010), pp. 193001.
[11]Daubechies, I., Ten Lectures on Wavelets, SIAM, Philadelphia, 1992.
[12]Debnath, L., Wavelet Transforms and Their Applications, Springer, 2002.
[13]Duffin, R. J. and Schaeffer, A. C., A class of nonharmonic Fourier series, Trans. Amer. Math. Soc., 72 (1952), pp. 341366.
[14]Kutyniok, G., Okoudjou, K. A., Philipp, F. and Tuley, E. K., Scalable frames, Linear Algebra Appl., 438 (2013), pp. 22252238.
[15]Li, D. F. and Sun, W. C., Expansion of frames to tight frames, Acta Mathematica Sinica, English Series, 25 (2009), pp. 287292.
[16]Mallat, S., A Wavelet Tour of Signal Processing-The Sparse Way (3nd Edition), Beijing, China Machine Press, 2010.
[17]Meyer, Y., Wavelets and Operators, Cambridge University Press, 1995.
[18]Paul, T. and Seip, K., Wavelets and quantum mechanics, wavelets and their applications, Jones and Bartlett, Boston, MA, 1992, pp. 303321.
[19]Prugovec˘ki, E., Quantum Mechanics in Hilbert Space, Academic Press, 1982.
[20]Renner, R. and Cirac, J. I., de Finetti representation theorem for infinite-dimensional quantum systems and applications to quantum cryptography, Phys. Rev. Lett., (102) 2009, pp. 110504–1–110504–4.
[21]Vernaz-Gris, P., Ketterer, A., Keller, A., Walborn, S., Coudreau, T. and Milman, P., Continuous discretization of infinite-dimensional Hilbert spaces, Phys. Rev. A, 89 (2014), pp. 052311–1–052311–13.
[22]Zhang, S. and Vourdas, A., Analytic representation of finite quantum systems, J. Phys. A Math. General, 37 (2004), pp. 83498363.

Keywords

Bessel Sequences and Its F-Scalability

  • Lei Liu (a1), Xianwei Zheng (a1), Jingwen Yan (a2) and Xiaodong Niu (a3)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed