Advertisement

A Note on Approximating Finite Hilbert Transform and Quadrature Formula

  • Fuat UstaEmail author
Article
  • 36 Downloads

Abstract

In this study, approximations for the finite Hilbert transform are given utilizing the fundamental integral identity for absolutely continuous mappings. Then, a numerical integration for this transform is obtained. Finally some numerical experiments have been presented.

Keywords

Finite Hilbert transform CPV (Cauchy Principal Value) Absolutely continuous mappings 

Mathematics Subject Classification

41A15 41A55 

Notes

References

  1. 1.
    King, F.W.: Hilbert Transforms, Volume 1, Encyclopedia of Mathematics and its Applications 124. Cambridge University Press, New York (2009)Google Scholar
  2. 2.
    Dragomir, N.M., Dragomir, S.S., Farrell, P.M., Baxter, G.W.: A quadrature rule for the finite Hilbert transform via trapezoid type inequalities. J. Appl. Math. Comput. 13(1–2), 67–84 (2003)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Dragomir, N.M., Dragomir, S.S., Farrell, P.: Approximating the finite Hilbert transform via trapezoid type inequalities. Comput. Math. Appl. 43(10–11), 1359–1369 (2002)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Liu, W.J., Lu, N.: Approximating the finite Hilbert Transform via Simpson type inequalities and applications, Politehnica University of Bucharest Scientific Bulletin-Series A-Applied. Math. Phys. 77(3), 107–122 (2015)MathSciNetGoogle Scholar
  5. 5.
    Dragomir, S.S.: Approximating the finite Hilbert transform via Ostrowski type inequalities for absolutely continuous functions. Bull. Korean Math. Soc. 39(4), 543–559 (2002)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Liu, W., Gao, X., Wen, Y.: Approximating the finite Hilbert transform via some companions of Ostrowskis inequalities. Bull. Malays. Math. Sci. Soc. 39(4), 1499–1513 (2016)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Usta, F.: Approximating the finite hilbert transform for absolutely continuous mappings and applications in numerical integration. Adv. Appl. Clifford Algebras 28, 78 (2018).  https://doi.org/10.1007/s00006-018-0898-z MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Usta, F.: On approximating the finite hilbert transform and applications in quadrature. Math. Methods Appl. Sci. (2018).  https://doi.org/10.1002/mma.5252 CrossRefzbMATHGoogle Scholar
  9. 9.
    Wang, S., Gao, X., Lu, N.: A quadrature formula in approximating the finite Hilbert transform via perturbed trapezoid type inequalities. J. Comput. Anal. Appl. 22(2), 239–246 (2017)MathSciNetGoogle Scholar
  10. 10.
    Wang, S., Lu, N., Gao, X.: A quadrature rule for the finite Hilbert transform via Simpson type inequalities and applications. J. Comput. Anal. Appl. 22(2), 229–238 (2017)MathSciNetGoogle Scholar
  11. 11.
    Liu, W., Gao, X.: Approximating the finite Hilbert transform via a companion of Ostrowski’s inequality for function of bounded variation and applications. Appl. Math. Comput. 247, 373–385 (2014)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematics Faculty of Science and ArtsDüzce UniversityDüzceTurkey

Personalised recommendations