Research PapersNo Access

Some results on wavelet scalograms

Saifallah Ghobber

Department of Mathematics, College of Education Al Majmaah, Majmaah University, P. O. Box 66, 11952 Al Majmaah, Kingdom of Saudi Arabia

LR11ES11 Analyse Mathématiques et Applications, Faculté des Sciences de Tunis, Université de Tunis El Manar, 2092 Tunis, Tunisie

E-mail Address: s.ghobber@mu.edu.sa

Search for more papers by this author

https://doi.org/10.1142/S0219691317500199Cited by:15 (Source: Crossref)

Abstract

Wavelets are a relatively recent development in applied mathematics. Motivated by the recent paper [On accumulated spectrograms, Trans. Amer. Math. Soc.368 (2016) 3629–3649] by Abreu, Gröchenig and Romero, we will prove in the present paper related results in the context of wavelets and time-scale analysis.

Keywords:

AMSC: 81S30, 94A12, 45P05, 42C25, 42C40

References

1. L. D. Abreu, Super-wavelets versus poly-Bergman spaces, Int. Equ. Oper. Theory 73 (2012) 177–193. Crossref, Web of Science, Google Scholar
2. L. D. Abreu, Superframes and polyanalytic wavelets, J. Fourier Anal. Appl. 23 (2017) 1–20. Crossref, Web of Science, Google Scholar
3. L. D. Abreu and J. E. Gilbert, Wavelet-type frames for the interval, Exp. Math. 32 (2014) 274–283. Crossref, Web of Science, Google Scholar
4. L. D. Abreu and K. Gröchenig, Banach Gabor frames with Hermite functions: Polyanalytic spaces from the Heisenberg group, Appl. Anal. 91 (2012) 1981–1997. Crossref, Web of Science, Google Scholar
5. L. D. Abreu, K. Gröchenig and J. L. Romero, On accumulated spectrograms, Trans. Amer. Math. Soc. 368 (2016) 3629–3649. Crossref, Web of Science, Google Scholar
6. P. S. Addison, J. N. Watson, G. R. Clegg, P. A. Steen and C. E. Robertson, Finding coordinated atrial activity during ventricular fibrilation using wavelet decomposition, analyzing surface ECGs with a new analysis technique to better understand cardiac death, IEEE Trans. Eng. Med. Biol. 21 (2002) 58–65. Crossref, Web of Science, Google Scholar
7. C. K. Chui, An introduction to wavelets, in Wavelet Analysis and Its Applications, Vol. 1 (Academic Press Inc., Boston, MA, 1992). Google Scholar
8. I. Daubechies, Ten lectures on wavelets, in CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 61 (Society for Industrial and Applied Mathematics, Philadelphia, PA, 1992). Google Scholar
9. I. Daubechies, The wavelet transform, time-frequency localization and signal analysis, IEEE Trans. Inform. Theory 36 (1990) 961–1005. Crossref, Web of Science, Google Scholar
10. M. Dörfler and J. L. Romero, Frames adapted to a phase-space cover, Constr. Approx. 39 (2014) 445–484. Crossref, Web of Science, Google Scholar
11. S. Mallat, A wavelet tour of signal processing, in The Sparse Way, With contributions from Gabriel Peyré, 3rd edn. (Elsevier/Academic Press, Amsterdam, 2009). Google Scholar
12. F. Riesz and B. Sz.-Nagy, Functional Analysis (Frederick Ungar Publishing Co., New York, 1955). Google Scholar
13. J. L. Romero, Surgery of spline-type and molecular frames, J. Fourier Anal. Appl. 17 (2011) 135–174. Crossref, Web of Science, Google Scholar
14. J. L. Romero, Characterization of coorbit spaces with phase-space covers, J. Funct. Anal. 262 (2012) 59–93. Crossref, Web of Science, Google Scholar
15. P. Sukiennik and J. T. Białasiewicz, Cross-correlation of bio-signals using continuous wavelet transform and genetic algorithm, J. Neurosci. Meth. 247 (2015) 13–22. Crossref, Web of Science, Google Scholar
16. E. Wilczok, New uncertainty principles for the continuous Gabor transform and the continuous wavelet transform, Doc. Math. 5 (2000) 201–226. Crossref, Google Scholar
17. M. W. Wong, Wavelet Transforms and Localization Operators (Birkhäuser, 2002). Crossref, Google Scholar