Skip to main content
SpringerLink
Log in

Continuous wavelet transform with the Shannon wavelet from the point of view of hyperbolic partial differential equations

Непрерывное вейвлет-преобразование с вейвлетом Шенпопа с точки зрения гиперболических дифференциальных уравнений в частных производных

  • Published:
Analysis Mathematica Aims and scope Submit manuscript

Abstract

We identify the result of the continuous wavelet transform with the difference of solutions of two hyperbolic partial differential equations, for which wavelet’s shift and scale are considered as independent variables on 2D plane. The characteristic property, which follows from the introduced representation is the fact that the transform’s values inside the triangle defined by two characteristics (a = const, b = const) and crossecting the slopped line on a scale-shift plane (a, b) are completely and uniquely defined by the values of the transform and its derivatives along the last mentioned line.

Резюме

Мы идентифицируем результат непрерывного веИвлет-преобразования с разностью решениИ двух гиперболических дифференциальных уравнении в частных производных, для которых сдвиг и масштаб рассматриваются как независимые переменные на двумернои плоскости. Характеристическое своиство, которое следует из введенного представление состоит в факте, что значение преобразования внутри треугольника, определенного двумя характеристиками (a = const, b = const) и пересекающеи их прямой на плоскости масштабов-свигов (а, б), полностью и однозначно определяется значениями преобразования и его производных вдоль указанных отрезков.

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. I. Ya. Novikov, M. A. Skopina, and V. Yu. Protasov, Wavelet Theory, Amer. Math. Soc. (Providence, RI, 2011).

    MATH  Google Scholar 

  2. A. Teolis, Computational Signal Processing with Wavelets, Birkhäuser (Boston, MA, 1998).

    MATH  Google Scholar 

  3. C. K. Chui and J. Wang, Shannon wavelet approach to sub-band coding, Int. J. Wavelets Multiresolut. Inf. Process., 1(2003), 233–242.

    Article  MATH  MathSciNet  Google Scholar 

  4. R. Pavlík, Binary psk/cpfsk and msk bandpass modulation identifier based on the complex Shannon wavelet transform, J. Electrical Engineering, 56(2005), 71–77.

    Google Scholar 

  5. S.-Y. Wang, T. Z. Aziz, J. F. Stein, and X. Liu, Time-frequency analysis of transient neuromuscular events: dynamic changes in activity of the subthalamic nucleus and forearm muscles related to the intermittent resting tremor, J. Neurosci. Meth., 145(2005), 151–158.

    Article  Google Scholar 

  6. A. Almanji and J.-Y. Chang, Feature extraction of surface electromyography signals with continuous wavelet entropy transform, Microsys. Technol., 17(2011), 1187–1196.

    Article  Google Scholar 

  7. E. B. Postnikov, Evaluation of a continuous wavelet transform by solving the Cauchy problem for a system of partial differential equations, Comp. Math. Math. Phys., 46(2006), 73–78.

    Article  MathSciNet  Google Scholar 

  8. E. B. Postnikov and A. Y. Loskutov, Wavelet analysis of fine-scale structures in the Saturnian B and C rings using data from the Cassini spacecraft, J. Exp. Theor. Phys., 104(2007), 417–422.

    Article  Google Scholar 

  9. M. V. Perel and M. S. Sidorenko, New physical wavelet ‘Gaussian wave packet’, J. Phys. A, 40(2007), 3441–3461.

    Article  MATH  MathSciNet  Google Scholar 

  10. C. S. Cho, S. W. Ha, J. C. Kim, T. H. Yoon, and K. G. Nam, Optoelectronic difference-of-gaussian wavelet transform system, Opt. Eng., 36(1997), 3471–3475.

    Article  Google Scholar 

  11. N. S. Koshlyakov, M. M. Smirnov, and E. B. Gliner, Differential Equations of Mathematical Physics, North-Holland Publishing Company (Amsterdam, 1964).

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Theoretical Physics, Kursk State University, Radishcheva st. 33, Kursk, 305000, Russia

    Eugene B. Postnikov

  2. Department of Mathematical Sciences, Indian Institute of Technology, Banaras Hindu University, Varanasi, India

    Vineet K. Singh

Authors
  1. Eugene B. Postnikov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Vineet K. Singh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Eugene B. Postnikov.

Additional information

The first author was supported by Grant No. 1391 of the Ministry of Education and Science of the Russian Federation within the basic part of research funding No. 2014/349 assigned to Kursk State University. The seciond author is grateful for the financial support of the BOYSCAST Fellowship 2010-2011 program of the Department of Science and Technology of the Government of India.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Postnikov, E.B., Singh, V.K. Continuous wavelet transform with the Shannon wavelet from the point of view of hyperbolic partial differential equations. Anal Math 41, 199–206 (2015). https://doi.org/10.1007/s10476-015-0206-2

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10476-015-0206-2

Keywords

Search

Navigation