Skip to main content
SpringerLink
Log in

Abelian Theorems for Quadratic-Phase Fourier Wavelet Transform

  • Scientific Research Paper
  • Published:
Proceedings of the National Academy of Sciences, India Section A: Physical Sciences Aims and scope Submit manuscript

Abstract

In this paper, we establish the Abelian theorems, which characterize the behaviour of the quadratic-phase Fourier wavelet transform (QPFWT) \(\Big (W^{\varOmega ^{a, b, c}_{d, e}}_{\psi }f\Big )\) \((\beta ,\alpha )\) as \(\alpha \rightarrow \infty \) to the behaviour to determine the function and tempered distribution \( \hat{f}(\omega ) \,as \,\omega \rightarrow 0\). These results are mentioned as initial and final value Abelian theorems. Some results on asymptotic orders of QPFWT are also 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. Castro LP, Minh LT, Tuan NM (2018) New convolutions for quadratic-phase Fourier integral operators and their applications. Mediterr J Math 15(1):1–17

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  Google Scholar 

  3. Pathak RS, Prasad A, Kumar M (2012) Fractional Fourier transform of tempered distributions and generalized pseudo-differential operator. J Pseudo-Differ Oper Appl 3(2):239–254

    Article  MathSciNet  MATH  Google Scholar 

  4. Prasad A, Manna S, Mahato A, Singh VK (2014) The generalized continuous wavelet transform associated with the fractional Fourier transform. J Comput Appl Math 259:183–194

    Article  MathSciNet  MATH  Google Scholar 

  5. Trigub RM, Belinsky ES (2004) Fourier analysis and approximation of functions. Kluwer Academic, Bostan

    Book  Google Scholar 

  6. Zayed AI (1998) Fractional Fourier transform of generalized functions. Integral Transforms Spec Funct 7:299–312

    Article  MathSciNet  MATH  Google Scholar 

  7. Prasad A, Sharma PB (2020) Quadratic phase Fourier wavelet transform. Math Methods Appl Sci 43(4):1953–1969

    Article  MathSciNet  MATH  Google Scholar 

  8. Daubechies I (2006) Ten lecture on wavelet, in CBMS-NSF regional conference series in applied mathematics. SIAM, Philadelphia, PA

    Google Scholar 

  9. Pathak RS (2009) The wavelet transform, vol 4. Atlantis press/Word scientific, Amsterdam

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  11. Pathak RS (2001) Abelian theorems for the wavelet transform, Wavelet and Allied Topics, cds. Jain PK et al. Narosa publishing House, New Delhi

  12. Prasad A, Kumar P (2017) Abelian theorems for fractional wavelet transform. Asian-Eur J Math 10(1):1750019

    Article  MathSciNet  MATH  Google Scholar 

  13. Rieder A (1991) The wavelet transform on Sobolev spaces and its approximation properties. Numer Math 58(8):875–894

    MathSciNet  MATH  Google Scholar 

  14. Zemanian AH (1965) Distribution theory and transform analysis. McGraw-Hill Book, New York

    MATH  Google Scholar 

  15. Zemanian AH (1966) Some Abelian theorems for the distributional Hankel and K-transformations. SIAM J Appl Math 14(6):1255–1265

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

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

    P. B. Sharma & A. Prasad

Authors
  1. P. B. Sharma
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. Prasad
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. 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

Sharma, P.B., Prasad, A. Abelian Theorems for Quadratic-Phase Fourier Wavelet Transform. Proc. Natl. Acad. Sci., India, Sect. A Phys. Sci. 93, 75–83 (2023). https://doi.org/10.1007/s40010-022-00790-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40010-022-00790-z

Keywords

Mathematics Subject Classification

Search

Navigation