SpringerLink
Log in

Construction Biorthogonal Wavelets and Frames

  • Chapter
  • First Online:
Construction of Wavelets Through Walsh Functions

Part of the book series: Industrial and Applied Mathematics ((INAMA))

  • 532 Accesses

Abstract

In this chapter, basic properties of biorthogonal wavelets on positive real lines and Vilenkin groups, frames on Cantor group, Parseval frames on Vilenkin group, and application of biorthogonal dyadic wavelets to image processing are presented and these results are discussed in more detail Farkov (Facta Univers (Nis) ser. Elec Eng 21: 309–325, 2008), Farkov, Maksimov, and Stroganov (Int. J. Wavelets Multiresolution Inf Process 9: 485–499, 2011), Farkov (J Math Sci 187: 22–34, 2012).

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

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 119.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 159.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free ship** worldwide - see info
Hardcover Book
USD 159.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free ship** worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Farkov, Yu. A. (2008). Multiresolution analysis and wavelets on vilenkin groups. Facta Univers. (Nis) ser.: Elec. Engineering, 21(3), 309–325.

    Google Scholar 

  2. Farkov, Yu. A, Maksimov, A. Yu., & Stroganov, S. A. (2011). On biorthogonal wavelets related to the Walsh functions. International Journal of Wavelets, Multiresolution and Information Processing, 9(3), 485–499.

    Google Scholar 

  3. Farkov, Yu. A. (2012). Examples of frames on the Cantor dyadic group. Journal of Mathematical Sciences, 187(1), 22–34.

    Article  MathSciNet  Google Scholar 

  4. Christensen, O. (2003). An introduction to frames and Riesz bases. Basel: Birkhaeuser.

    Google Scholar 

  5. Golubov, B. I., Efimov, A. V., & Skvortsov, V. A. (2008). Walsh series and transforms (English Translation of 1st ed.). Moscow: Urss; Dordrecht: Kluwer (1991).

    Google Scholar 

  6. Schipp, F., Wade, W. R., & Simon, P. (1990). Walsh series. Bristol: Adam Hilger.

    Google Scholar 

  7. Lang, W. C. (1996). Orthogonal wavelets on the Cantor dyadic group. SIAM Journal on Mathematical Analysis, 27(1), 305–312.

    Article  MathSciNet  Google Scholar 

  8. Lang, W. C. (1998). Fractal multiwavelets related to the Cantor dyadic group. International Journal of Mathematics and Mathematical Sciences, 21, 307–317.

    Article  MathSciNet  Google Scholar 

  9. Lang, W. C. (1998). Wavelet analysis on the Cantor dyadic group. Houston Journal of Mathematics, 24, 533–544.

    MathSciNet  MATH  Google Scholar 

  10. Farkov, Yu. A. (2005). Orthogonal p-wavelets on \(R^{+}\). In Proceedings of International Conference Wavelets and Splines (St. Petersburg, Russia, July 3–8) (p. 426). St. Petersberg: St. Petersberg University Press.

    Google Scholar 

  11. Farkov, Yu. A. (2005). Orthogonal Wavelets with compact support on locally compact abelian groups. Izv. Ross. Akad. Nauk Ser. Mat. 69(3), 193–220. English translation, Izvestia: Mathematics 69(3), 623–650.

    Article  MathSciNet  Google Scholar 

  12. Farkov, Yu. A. & Protasov V. Yu. (2006). Dyadic wavelets and refinable functions on a half line. Mat. Sbornik, 197(10), 129–160. English translation, Sbornik: Mathematics, 197, 1529–1558.

    Google Scholar 

  13. Farkov, Yu. A. (2009). Biorthogonal wavelets on Vilenkin groups. Tr. Mat. Inst. Steklova 265(1), 110–124. English translation, Proceedings of the Steklov Institute of Mathematics, 265(1), 101–114.

    Google Scholar 

  14. Novikov, I. Ya., Protasov, V. Yu., & Skopina, M. A. (2011). Wavelet theory (Moscow, 2006). Providence: AMS.

    Google Scholar 

  15. Daubechies, I. (1992). Ten lectures on wavelets. Philadelphia: SIAM.

    Book  Google Scholar 

  16. Farkov, Yu. A., & Rodionov, E. A. (2009). Estimates of the smoothness of dyadic orthogonal wavelets of Daubechies type. Mathematical Notes, 86(3), 407–421.

    Google Scholar 

  17. Welstead, S. (2000). Fractal and wavelet image compression techniques. Bellingham: SPIE Optical Engineering Press.

    Google Scholar 

  18. Hereford, J., Roach, D. W., & Pigford, R. (2003). Image compression using parameterized wavelet with feedback. Proceeding SPIE, 5102, 267–277

    Google Scholar 

  19. Kozyrev, S. V. (2002). Wavelet analysis as a p-adic spectral analysis. Izvestiya: Mathematics, (66), 367–376.

    Article  MathSciNet  Google Scholar 

  20. Benedetto, J. J., & Benedetto, R. L. (2009). The construction of wavelet sets, wavelets and multiscale analysis. In Theory and Applications. Selected papers based on the presentations at the International conference on Wavelets: Twenty years of Wavelets. Cohen, J. et al. (ed.) De Paul University, Chicago, IL, USA, May 15–17, New York: Springer (Alied And Numerical Harmonic Analysis), (pp. 17–56).

    Google Scholar 

  21. Agaev, G. H., Vilemkin, N. Ya., Dzhafarli, G. M., & Rubinstein, A. I. (1981). Multiplicative systems of functions and analysis on 0 dimensional groups. Baku: ELM. [In Russian].

    Google Scholar 

  22. Pontryagin, L. S. (1939). Topological groups. New Jersey: University Press Princiton.

    MATH  Google Scholar 

  23. Edwards, R. E. (1982). Fourier series: A modern introduction (Vol. 2). Berlin: Springer.

    Book  Google Scholar 

  24. Benedetto, J. J., & Benedetto, R. L. (2004). A wavelet theory for local fields and related groups. Journal of Geometric Analysis, 14, 423–456.

    Article  MathSciNet  Google Scholar 

  25. Farkov, Yu. A. (2007). Orthogonal wavelets on direct products of cyclic groups. Matematicheskie Zametki, 82(6), 934–952. English Translation, Mathematical Notes, 82(6), 843–859.

    Google Scholar 

  26. Protasov, V. Yu., & Farkov, Yu A. (2006). Dyadic wavelet and refinable functions on a half line. Sboinik: Mathematics, 197(10), 1529–1558.

    MATH  Google Scholar 

  27. Chui, C. K., & Mhaskar, H. N. (1993). On trigonometric wavelets. Constructive Approximation, 9, 167–190.

    Article  MathSciNet  Google Scholar 

  28. Shah, F. A. (2012). Biorthogonal \(p\)-wavelet packets related to the Walsh polynomials. Journal of Classical Analysis, 1(2), 135–146.

    Article  MathSciNet  Google Scholar 

  29. Farkov, Yu. A. (2010). Wavelets and frames based on Walsh-Dirichlet type kernels. Communications in Mathematics and Applications, 1, 27–46.

    Google Scholar 

  30. Siddiqi, A. H. (1978). Walsh function. Aligarh: AMU.

    Google Scholar 

  31. Farkov, Yu. A. (2009). On wavelets related to Walsh series. Journal of Approximation Theory, 161, 259–279.

    Article  MathSciNet  Google Scholar 

  32. Farkov, Yu. A, & Rodionov, E. A. (2012). Nonstationary wavelets related to the Walsh functions. American Journal of Computational Mathematics, 2, 82–87.

    Google Scholar 

  33. Farkov, Yu. A, & Stroganov, S. A. (2011). The use of discrete dyadic wavelets in image processing. Russian Mathematics (Iz. Yuz), 55(7), 47–55. Original Russian Text Published in Izvestiya. Uchebnykh Zavednic, Mathematica, (7), 57–66.

    Google Scholar 

  34. Farkov, Yu. A. (2015). Construction of MRA- based wavelets and frames in Walsh analysis. Poincare Journal of Analysis and Applications, 2015(2), Special Issue (IWWFA-II, Delhi), 13–36.

    Google Scholar 

  35. Farkov, Yu. A. (2014). Wavelet expansions on the Cantor group. Mathematical Notes, 96(6), 996–1007.

    Article  MathSciNet  Google Scholar 

  36. Daubechies, I. (1988). Orthonormal bases of compactly supported wavelets. Communications on Pure and Applied Mathematics, 41, 909–996.

    Article  MathSciNet  Google Scholar 

  37. Farkov, Yu. A., & Rodionov, E. A. (2011). Algorithms for wavelet construction on Vilenkin groups. P-Adic Numbers, Ultrametric Analysis, and Applications, 3(3), 181–195.

    Google Scholar 

  38. Farkov, Yu. A., Lebedeva, E. A., & Skopina, M. A. (2015). Wavelet frames on Vilenkin groups and their approximation properties. International Journal of Wavelets, Multiresolution and Information Processing, 13(5). https://doi.org/10.1142/5021.

  39. Farkov, Yu. A. (2016). MRA-Based Wavelet Frames in Walsh Analysis, Conference on Harmonic Analysis and Approximation Theory, (CRM, Bellaterria, Spain).

    Google Scholar 

  40. Dong, B. & Shen, Z. (2013). Framelets: MRA-based wavelet frames and applications, in Mathematics in image processing / Hogkai Zhao editor.- (IAS/Park City mathematics series) 7-208.

    Google Scholar 

  41. Farkov, Yu. A. Wavelet frames related to Walsh functions. European Journal of Mathematics. https://doi.org/10.1007/s40879-018-0220-6. (in press).

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Applied Information Technology, School of Public Policy, Russian Presidential Academy of National Economy and Public Administration (RANEPA), Moscow, Russia

    Yu. A. Farkov

  2. Department of Mathematics, Guru Nanak Dev University, Amritsar, Punjab, India

    Pammy Manchanda

  3. School of Basic Sciences and Research and Centre for Advanced Research in Applied Mathematics and Physics, Sharda University, Greater Noida, Uttar Pradesh, India

    Abul Hasan Siddiqi

Authors
  1. Yu. A. Farkov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pammy Manchanda
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Abul Hasan Siddiqi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yu. A. Farkov .

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Singapore Pte Ltd.

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Farkov, Y.A., Manchanda, P., Siddiqi, A.H. (2019). Construction Biorthogonal Wavelets and Frames. In: Construction of Wavelets Through Walsh Functions. Industrial and Applied Mathematics. Springer, Singapore. https://doi.org/10.1007/978-981-13-6370-2_7

Download citation

Publish with us

Policies and ethics

Search

Navigation