Skip to main content

Generalized Finite Hilbert Transform and Some Basic Inequalities

Abstract

In this paper we consider a generalized finite Hilbert transform of complex valued functions and establish some basic inequalities for several particular classes of interest. Applications for some particular instances of finite Hilbert transforms are given as well.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. N.M. Dragomir, S.S. Dragomir, P.M. Farrell, Some inequalities for the finite Hilbert transform, in Inequality Theory and Applications, Nova Science Publishers, vol. I (Huntington, NY, 2001), pp. 113–122

    MATH  Google Scholar 

  2. N.M. Dragomir, S.S. Dragomir, P.M. Farrell, Approximating the finite Hilbert transform via trapezoid type inequalities. Comput. Math. Appl. 43(10–11), 1359–1369 (2002)

    CrossRef  MathSciNet  Google Scholar 

  3. N.M. Dragomir, S.S. Dragomir, P.M. Farrell, G.W. Baxter, On some new estimates of the finite Hilbert transform. Libertas Math. 22, 65–75 (2002)

    MathSciNet  MATH  Google Scholar 

  4. N.M. Dragomir, S.S. Dragomir, P.M. Farrell, G.W. Baxter, A quadrature rule for the finite Hilbert transform via trapezoid type inequalities. J. Appl. Math. Comput. 13(1–2), 67–84 (2003)

    CrossRef  MathSciNet  Google Scholar 

  5. N.M. Dragomir, S.S. Dragomir, P.M. Farrell, G.W. Baxter, A quadrature rule for the finite Hilbert transform via midpoint type inequalities, in Fixed Point Theory and Applications, Nova Science Publishers, vol. 5, (Hauppauge, NY, 2004), pp. 11–22

    Google Scholar 

  6. S.S. Dragomir, Approximating the finite Hilbert transform via an Ostrowski type inequality for functions of bounded variation. J. Inequal. Pure Appl. Math. 3(4), Article 51, 19 (2002)

    Google Scholar 

  7. S.S. Dragomir, Approximating the finite Hilbert transform via Ostrowski type inequalities for absolutely continuous functions. Bull. Korean Math. Soc. 39(4), 543–559 (2002)

    CrossRef  MathSciNet  Google Scholar 

  8. S.S. Dragomir, Inequalities for the Hilbert transform of functions whose derivatives are convex. J. Korean Math. Soc. 39(5), 709–729 (2002)

    CrossRef  MathSciNet  Google Scholar 

  9. S.S. Dragomir, Some inequalities for the finite Hilbert transform of a product. Commun. Korean Math. Soc. 18(1), 39–57 (2003)

    CrossRef  MathSciNet  Google Scholar 

  10. S.S. Dragomir, Sharp error bounds of a quadrature rule with one multiple node for the finite Hilbert transform in some classes of continuous differentiable functions. Taiwanese J. Math. 9(1), 95–109 (2005)

    CrossRef  MathSciNet  Google Scholar 

  11. S.S. Dragomir, The perturbed median principle for integral inequalities with applications, in Nonlinear Analysis and Variational Problems, Springer Optimization and Its Applications, vol. 35 (Springer, New York, 2010), pp. 53–63

    Google Scholar 

  12. S.S. Dragomir, Inequalities and approximations for the Finite Hilbert transform: a survey of recent results, Preprint RGMIA Res. Rep. Coll.21, Article 30, 560 (2018). http://rgmia.org/papers/v21/v21a30.pdf

  13. F.D. Gakhov, Boundary Value Problems (English translation) (Pergamon Press, Oxford, 1966)

    MATH  Google Scholar 

  14. W. Liu, X. Gao, 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)

    MathSciNet  MATH  Google Scholar 

  15. W. Liu, X. Gao, Y. Wen, Approximating the finite Hilberttransform via some companions of Ostrowski’s inequalities. Bull. Malays. Math. Sci. Soc. 39(4), 1499–1513 (2016)

    CrossRef  MathSciNet  Google Scholar 

  16. W. Liu, N. Lu, Approximating the finite Hilbert transform via Simpson type inequalities and applications. Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 77(3), 107–122 (2015)

    MathSciNet  MATH  Google Scholar 

  17. S.G. Mikhlin, S. Prössdorf, Singular Integral Operators (English translation) (Springer Verlag, Berlin, 1986)

    CrossRef  Google Scholar 

  18. S. Wang, X. Gao, N. Lu, A quadrature formula in approximating the finite Hilbert transform via perturbed trapezoid type inequalities. J. Comput. Anal. Appl. 22(2), 239–246 (2017)

    MathSciNet  Google Scholar 

  19. S. Wang, N. Lu, X. Gao, A quadrature rule for the finite Hilberttransform via Simpson type inequalities and applications. J. Comput. Anal. Appl. 22(2), 229–238 (2017)

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Silvestru Sever Dragomir .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Dragomir, S.S. (2019). Generalized Finite Hilbert Transform and Some Basic Inequalities. In: Anastassiou, G., Rassias, J. (eds) Frontiers in Functional Equations and Analytic Inequalities. Springer, Cham. https://doi.org/10.1007/978-3-030-28950-8_27

Download citation