Skip to main content
SpringerLink
Log in

Superframes and Polyanalytic Wavelets

  • Published:
Journal of Fourier Analysis and Applications Aims and scope Submit manuscript

Abstract

We construct a polyanalytic extension of analytic (Poisson) wavelet frames, leading to a new system of wavelet superframes. Superframes have been introduced in signal analysis as a tool for the multiplexing of signals—encoding several signals as a single one with the purpose of sharing a communication channel. In this paper, a new system of vector-valued wavelets is constructed by selecting as elements of the analyzing vector the first n elements from an explicit orthogonal basis of the space of admissible functions depending on a parameter \(\alpha \). We show that if the resulting discrete affine system indexed by the set \(\Gamma (a,b)=\{a^{m}bk,a^{m}\}_{k,m\in \mathbb {Z} }\) is a wavelet superframe, then the estimate

$$\begin{aligned} b\log a<\frac{2\pi }{n+\alpha } \end{aligned}$$

holds. This is proved by defining a polyanalytic Bergman transform, a unitary map which rephrases the problem in terms of sampling in polyanalytic Bergman spaces of the upper half-plane. Besides the applications in multiplexing, polyanalytic superframes lead to discrete counterparts of multitapered wavelet representations used in spectrum estimation and in high resolution time–frequency representation algorithms.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abreu, L.D.: Sampling and interpolation in Bargmann–Fock spaces of polyanalytic functions. Appl. Comput. Harmon. Anal. 29, 287–302 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  2. Abreu, L.D.: Wavelet frames with Laguerre functions. C. R. Acad. Sci. Paris Ser. 349, 255–258 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Abreu, L.D.: Super-wavelets versus poly-Bergman spaces. Integr. Equ. Oper. Theory 73, 177–193 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  4. Abreu, L.D., Balazs, P., de Gosson, M., Mouhayn, Z.: Discrete coherent states for higher Landau levels. Ann. Phys. 363, 337–353 (2015)

    Article  MathSciNet  Google Scholar 

  5. Abreu, L.D., Dörfler, M.: An inverse problem for localization operators. Inverse Probl. 28, 115001 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. Abreu, L.D., Gröchenig, K.: Banach Gabor frames with Hermite functions: polyanalytic spaces from the Heisenberg group. Appl. Anal. 91, 1981–1997 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  7. Abreu, L.D., Feichtinger, H.G.: Function spaces of polyanalytic functions. In: Alexander Vasil’ev (ed.) Harmonic and Complex Analysis and Its Applications. Trends in Mathematics, pp. 1–38. Birkhäuser/Springer, Cham (2014)

  8. Abreu, L.D., Gröchenig, K., Romero, J.L.: On accumulated spectrograms. Trans. Am. Math. Soc. 368, 3629–3649 (2016)

  9. Ascensi, G., Bruna, J.: Model space results for the Gabor and Wavelet transforms. IEEE Trans. Inform. Theory 55, 2250–2259 (2009)

    Article  MathSciNet  Google Scholar 

  10. Balan, R.: Multiplexing of signals using superframes. In: SPIE Wavelets Applications, vol. 4119 of Signal and Image processing XIII, pp. 118–129 (2000)

  11. Balan, R.: Density and redundancy of the noncoherent Weyl-Heisenberg superframes. Contemp. Math. 247, 29–41 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  12. Bhatt, G., Johnson, B.D., Weber, E.: Orthogonal wavelet frames and vector-valued wavelet transforms. Appl. Comput. Harmon. Anal. 23, 215–234 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  13. Bildea, S., Dutkay, D.E., Picioroaga, G.: MRA super-wavelets. N. Y. J. Math. 11, 1–19 (2005)

    MathSciNet  MATH  Google Scholar 

  14. Comtet, A.: On the Landau levels on the hyperbolic plane. Ann. Phys. 173, 185–209 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  15. Daubechies, I.: Ten Lectures on Wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics (1992)

  16. Daubechies, I., Paul, T.: Time-frequency localisation operators—a geometric phase space approach. II. The use of dilations. Inverse Probl. 4, 661–680 (1988)

  17. Daubehies, I., Wang, Y.G., Wu, H.-T.: ConceFT: concentration of frequency and time via a multitapered synchrosqueezed transform. Philos Trans A Math Phys Eng Sci. 374(2065), 20150193 (2016). doi:10.1098/rsta.2015.0193

  18. de Gosson, M.: Spectral properties of a class of generalized Landau operators. Commun. Part. Differ. Equ. 33, 2096–2104 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. Dutkay, D.E., Jorgensen, P.: Oversampling generates super-wavelets. Proc. Am. Math. Soc. 135, 2219–2227 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  20. Feichtinger, H.G.: On a new Segal algebra, Monatsh. Math. 92 (1981), no. 4, 269–289. Gabor analysis and algorithms, 233–266, Appl. Numer. Harmon. Anal., Birkhäuser Boston, Boston (1998)

  21. Führ, H.: Simultaneous estimates for vector-valued Gabor frames of Hermite functions. Adv. Comput. Math. 29, 357–373 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  22. Gradshtein, I.S., Ryzhik, I.M.: Tables of Integrals, Series and Products, 7th edn. Academic Press, Elsevier, San Diego (2007)

    Google Scholar 

  23. Gröchenig, K., Lyubarskii, Y.: Gabor (Super) frames with Hermite functions. Math. Ann. 345, 267–286 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  24. Gu, Q., Han, D.: Super-wavelets and decomposable wavelet frames. J. Fourier Anal. Appl. 11(6), 683–696 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  25. Haimi, A., Hedenmalm, H.: The polyanalytic Ginibre ensembles. J. Stat. Phys. 153(1), 10–47 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  26. Han, D., Larson, D.R.: Frames, bases and group representations. Mem. Am. Math. Soc. 147(697) (2000)

  27. Hartmann, M.M., Matz, G., Schafhuber, D.: Wireless multicarrier communications via multipulse Gabor Riesz bases. EURASIP J. Appl. Signal Process. 23818, 1–15 (2006)

    Article  MATH  Google Scholar 

  28. Haykin, S.: Cognitive radio: brain-empowered wireless communications. IEEE J. Sel. Areas Commun. 23(2), 201–220 (2005)

    Article  Google Scholar 

  29. Heil, C., Kutyniok, G.: Density of weighted wavelet frames. J. Geom. Anal. 13, 479–493 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  30. Hutník, O.: A note on wavelet subspaces. Monatsh. Math. 160, 59–72 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  31. Hutník, O.: On Toeplitz-type operators related to wavelets. Integr. Equ. Oper. Theory 63, 29–46 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  32. Hutník, O.: Wavelets from Laguerre polynomials and Toeplitz-type operators. Integr. Equ. Oper. Theory 71, 357–388 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  33. Kutyniok, G.: Affine Density in Wavelet Analysis, Lecture Notes in Mathematics 1914, Springer, Berlin (2007)

  34. Mouayn, Z.: Characterization of hyperbolic Landau states by coherent state transforms. J. Phys. A Math. Gen. 36, 8071–8076 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  35. Olhede, S.C., Walden, A.T.: Generalized Morse wavelets. IEEE Trans. Signal. Process. 50(11), 2661–2671 (2002)

    Article  MathSciNet  Google Scholar 

  36. Omer, H., Torresani, B.: Time-frequency and time-scale analysis of deformed stationary processes, with applications to non-stationary sound modelling (2014) \(<\)hal-01094835\(>\)

  37. Patterson, S.J.: The laplacian operator on a Riemann surface. Compos. Math. 31, 227–259 (1975)

    MathSciNet  MATH  Google Scholar 

  38. Ricaud, B., Torresani, B.: A survey of uncertainty principles and some signal processing applications. Adv. Comput. Math. 40(3), 629–650 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  39. Seip, K.: Regular sets of sampling and interpolation for weighted Bergman spaces. Proc. Am. Math. Soc. 117, 213–220 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  40. Thomson, D.J.: Spectrum estimation and harmonic analysis. Proc. IEEE 70, 1055–1095 (1982)

    Article  Google Scholar 

  41. Vasilevski, N.L.: On the structure of Bergman and poly-Bergman spaces. Integr. Equ. Oper. Theory 33, 471–488 (1999)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

I would like to thank Yurii Lyubarskii for several discussions on the topic and to Georg Tauboeck for pointing out the paper [27] and the potential applications in wireless multicarrier communications. I also want to thank both reviewers for the careful reading of the manuscript, leading to several corrections, improvements on the readability and also for suggesting explicit descriptions of the potential applications in other disciplines, which have been incorporated in the introduction and last section of the paper. L.D. Abreu was supported by Austrian Science Foundation (FWF) START-project FLAME (‘Frames and Linear Operators for Acoustical Modeling and Parameter Estimation’; Y 551–N13).

Author information

Authors and Affiliations

  1. Acoustics Research Institute, Austrian Academy of Sciences, Wohllebengasse 12-14, 1040, Vienna, Austria

    Luís Daniel Abreu

Authors
  1. Luís Daniel Abreu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Luís Daniel Abreu.

Additional information

Communicated by Yura Lyubarskii.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Abreu, L.D. Superframes and Polyanalytic Wavelets. J Fourier Anal Appl 23, 1–20 (2017). https://doi.org/10.1007/s00041-015-9448-4

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00041-015-9448-4

Keywords

Search

Navigation