Skip to main content
SpringerLink
Log in

Abelian theorems and Calderón’s reproducing formula for linear canonical wavelet transform

  • Published:
Journal of Pseudo-Differential Operators and Applications Aims and scope Submit manuscript

Abstract

In this paper, the initial and final value Abelian theorems for linear canonical wavelet transform of function and tempered distributions are studied. Some basic properties and the Calderón’s reproducing formula for linear canonical wavelet transform are also discussed. Further, using the Mexican hat wavelet function, an application of Abelian theorems to LCWT is investigated.

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. Ahirrao, B.R., More, S.V.: Abelian theorems for one sided Laplace Hardy transformations. Indian J. Pure Appl. Math. 18(1), 50–54 (1987)

    MathSciNet  MATH  Google Scholar 

  2. Alieva, T., Bastiaans, M.J.: Properties of the linear canonical integral transformation. J. Opt. Soc. Amer. A. 24, 3658–3665 (2007)

    Article  Google Scholar 

  3. Almeida, L.B.: The fractional Fourier transform and time frequency representations. IEEE Trans. Signal Process. 42(11), 3084–3091 (1994)

    Article  Google Scholar 

  4. Calderón’s, A.P.: Intermediate spaces and interpolation, the complex method. Stud. Math. 24, 113–190 (1964)

    Article  MathSciNet  Google Scholar 

  5. Chen, L., Zhao, D.: Optical image encryption based fractional wavelet transform. Opt. Commun. 254, 361–367 (2005)

    Article  Google Scholar 

  6. Chui, C.K.: An Introduction to Wavelets. Academic Press, New York (1992)

    MATH  Google Scholar 

  7. Collins, S.A.: Lens-system diffraction integral written in terms of matrix optics. J. Opt. Soc. Am. 60, 1168–1177 (1970)

    Article  Google Scholar 

  8. Dange, S., Choudhary, M.S.: Distributional Abelian theorems for the generalized Stieltjes transform. J. Math. Anal. Appl. 128, 125–137 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  9. Daubechies, I.: Ten Lectures on Wavelets, (BMS-NSP) Regional Conference Series in Applied Mathematics. SIAM Publ. Philadelphia, PA (2006)

    Google Scholar 

  10. Debnath, L.: Wavelet Transforms and Their Applications. Birkhäuser, Boston (2002)

    Book  MATH  Google Scholar 

  11. Dou, J., He, Q., Peng, Y., Sun, Q., Liu, S., Liu, Z.: A convolution-based fractional transform. Opt. Quant. Electron. 48(8), 407 (2016)

    Article  Google Scholar 

  12. Dynkin, E.M.: Methods of the theory of singular integrals little wood-Paley theory and its applications. In: Khavin, V. P., Nikal’ski, N. K. (eds.) Commutative Harmonic Analysis IV, vol. 42, pp. 97–194. Springer, Berlin (Encycl. Sci. Math.) (1992)

  13. Folland, G.B., Stein, E.M.: Hardy Spaces on Homogeneous Groups. Princeton University Press, Princeton (1982)

    MATH  Google Scholar 

  14. Frazier, M., Jawerth, B., Weiss, G.: Littlewood-Paley Theory and the Study of Function Spaces, CBMS Regional Conference Series in Mathematics, vol. 79. American Mathematical Society, Providence, Rhode Island (1991)

  15. Gelfand, I.M., Shilov, G.E.: Generalized Function II. Academic press, New York (1968)

    Google Scholar 

  16. Guo, Y., Li, B.Z.: The linear canonical wavelet transform on some function spaces. Int. J. wavelets Multiresolut. Inf. Process. 16, 1850010 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  17. Healy, J.J., Kutay, M.M., Ozaktas, H.M., Sheridan, J.T.: Linear Canonical Transform: Theory and Applications, vol. 198. Springer, New York (2016)

    Book  MATH  Google Scholar 

  18. Herson, D.L.J., Heywood, P.: On the range of some fractional integrals. J. Lond. Math. Soc. 8(4), 607–614 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  19. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, vol. 204. Elsevier Amsterdam (2006)

  20. Kou, K., Xu, R.: Windowed linear canonical transform and its applications. Signal Process. 92, 179–188 (2012)

    Article  Google Scholar 

  21. Liu, Z., Chen, D., Ma, J., Wei, S., Zhang, Y., Dai, J., Liu, S.: Fast algorithm of discrete gyrator transform based on convolution operation. Optik 122(10), 864–867 (2011)

    Article  Google Scholar 

  22. Mendlovic, D., Zalevsky, Z., Mas, D., Garcia, J., Ferreira, C.: Fractional wavelet transform. Appl. Opt. 36(20), 4801–4806 (1997)

    Article  Google Scholar 

  23. Moshinsky, M., Quesne, C.: Linear canonical transformations and their unitary representation. J. Maths. Phys. 12, 1772–1783 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  24. Nefzi, B., Brahim, K.: Calderón’s reproducing formula and uncertainty principle for the continuous wavelet transform associated with the q-Bessel operator. J. Pseudo-Differ. Oper. Appl. 6(3), 495–522 (2018)

    Article  MATH  Google Scholar 

  25. Ozaktas, H.M., Zelevsks, Z., Kutay, M.A.: Fractional Fourier Transform with Applications in Optics and Signal Processing. Wiley, Hoboken (2001)

    Google Scholar 

  26. Pathak, R.S.: Continuity and inversion of the wavelet transform. Integral Transforms Spec. Funct. 6(1–4), 85–93 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  27. Pathak, R.S.: Abelian theorems for the wavelet transform. In: Jain, P.K., et al. (eds.) Wavelets and Allied Topics. Narosa Publishing House, New Delhi (2001)

  28. Pathak, R.S.: The Wavelet Transform, vol. 6. Atlantis Press World Scientific, Paris (2009)

    Book  MATH  Google Scholar 

  29. Pathak, R.S., Pandey, G.: Calderón’s reproducing formula for Hankel convolution. Int. J. Math. Math. Sci. 2006, 024217 (2006)

    Article  MATH  Google Scholar 

  30. Pei, S.C., Ding, J.J.: Relation between fractional operations and time-frequency distributions and their applications. IEEE Trans. Signal Process. 49, 1638–1655 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  31. Pinsky, M.A.: Integrability of the continuum wavelet kernel. Proc. Amer. Math. Soc. 132(6), 1729–1737 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  32. Prasad, A., Kumar, P.: Abelians theorem for fractional wavelet transform. Asian-Eur. J. Math. 10(1), 1750019 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  33. Prasad, A., Kumar, P.: Composition of continuous fractional wavelet transforms. Natl. Acad. Sci. Lett. 39(2), 115–120 (2016)

    Article  MathSciNet  Google Scholar 

  34. Prasad, A., Kumar, P.: The continuous fractional wavelet transform on generalized weighted Sobolev spaces. Asian-Eur. J. Math. 8(3), 1550054 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  35. Prasad, A., Kumar, T.: Canonical Hankel wavelet transformation and Calderón’s reproducing formula. Filomat 32(8), 2735–2743 (2018)

    Article  MathSciNet  Google Scholar 

  36. Prasad, A., Ansari, Z.A.: Continuous wavelet transform involving linear canonical transform. Natl. Acad. Sci. Lett. 42(4), 337–344 (2019)

    Article  MathSciNet  Google Scholar 

  37. Prasad, A., Kumar, M.: Product of two generalized pseudo-differential operators involving fractional Fourier transform. J. Pseudo-Differ. Oper. Appl. 2(3), 355–365 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  38. Prasad, A., Ansari, Z.A.: Approximation of linear canonical wavelet transform on the generalized Sobolev spaces. J. Pseudo-Differ. Oper. Appl. 10(4), 855–881 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  39. Prasad, A., Manna, S., Mahato, A., Singh, V.K.: The generalized continuous wavelet transform associated with the fractional Fourier transform. J. Comput. Appl. Math. 259, 660–671 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  40. Schwartz, L.: Theorie des distributions. Hermann, Peris (1978)

    MATH  Google Scholar 

  41. Shi, J., Zhang, N., Liu, X.: A novel fractional wavelet transform and its applications. Sci. China Inf. Sci. 55(6), 1270–1279 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  42. Srivastava, H.M., González, B.J., Negrín, E.R.: A new class of Abelian theorems for the Mehler–Fock transforms. Russ. J. Math. Phys. 24, 124–126 (2017)

  43. Srivastava, H.M., Upadhyay, S.K., Khatterwani, K.A.: A family of pseudo-differential operators on the Schwartz space associated with the fractional Fourier transform. Russian J. Math. Phys. 24(4), 534–543 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  44. Upadhyay, S.K., Tripathi, A.: Calderón’s reproducing formula for watson wavelet transform. Indian J. Pure Appl. Math. 46(3), 269–277 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  45. Upadhyay, S.K., Singh, R.: Integrability of the continuum Bessel wavelet kernel. Int. J. Wavelets Multiresolut. Inf. Process. 13, 1550032 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  46. Wang, J., Wang, Y., Wang, W., Ren, S.: Discrete linear canonical wavelet transform and its applications. EURASIP J. Adv. Signal Process. 2008, 29 (2018)

  47. Wei, D., Li, Y.M.: Generalized wavelet transform based on the convolution operator in the linear canonical transform domain. Optik 125(16), 4491–4496 (2014)

    Article  Google Scholar 

  48. Zemanian, A.H.: Distribution Theory and Transform Analysis. McGraw-Hill Book, New York (1965)

    MATH  Google Scholar 

Download references

Acknowledgements

Authors are very thankful to the anonymous reviewer for his valuable and constructive comments and suggestions.

Author information

Authors and Affiliations

  1. Department of Mathematics and Computing, Indian Institute of Technology (Indian School of Mines), Dhanbad, 826004, India

    Z. A. Ansari & Akhilesh Prasad

Authors
  1. Z. A. Ansari
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Akhilesh Prasad
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Akhilesh Prasad.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ansari, Z.A., Prasad, A. Abelian theorems and Calderón’s reproducing formula for linear canonical wavelet transform. J. Pseudo-Differ. Oper. Appl. 12, 4 (2021). https://doi.org/10.1007/s11868-021-00373-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11868-021-00373-3

Keywords

Mathematics Subject Classification

Search

Navigation