Skip to main content
SpringerLink
Log in

Generalized Shift Operators Generated by Convolutions of Integral Transforms

  • Conference paper
Current Trends in Analysis and Its Applications

Part of the book series: Trends in Mathematics ((RESPERSP))

Abstract

In this work we discuss a problem of the equivalence of two main approaches to introducing of generalized convolution operators. The first of them is based on the constructing of a generalized shift operator. The idea of the second approach is based on the works by Valentin Kakichev. On this problem we demonstrate on the examples of classical and nonclassical convolution constructions of integral transforms. In particular, we consider the shift operators defined by the convolutions for Hankel integral transform with the function j ν (xt)=(2xt)ν Γ(ν+1)J ν (xt) in the kernel. Here J ν (xt) is the Bessel function of the first kind of order ν, \(\operatorname{Re} \nu>-1/2\).

This work was completed with the support of the Strategic Development Program of Novgorod State University.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight 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

Institutional subscriptions

References

  1. J. Delsarte, Sur une extensions de la formule de Taylor. J. Math. Pures Appl. 17(3), 213–231 (1938)

    Google Scholar 

  2. J. Delsarte, Une extension nouvelle de la theorie de foncion periodiques de Bohr. Acta Math. 69, 259–317 (1939)

    Article  MathSciNet  Google Scholar 

  3. J. Delsarte, Hypergroupes et operateuers de permutation et de transmutation, in Colloques Internat., Nancy, (1956), pp. 29–44

    Google Scholar 

  4. A. Povzner, On differential equations of Sturm–Liouville type on a half-axis. Mat. Sb. 23(65), 1 (1948), 3–52 (in Russian). Translation no. 5, Am. Math. Soc. (1950)

    MathSciNet  Google Scholar 

  5. B.M. Levitan, The application of generalized displacement operators to linear differential equations of the second order. Usp. Mat. Nauk 4, 1 (1949), 3–112 (in Russian). Translation no. 59, Am. Math. Soc. (1950)

    MathSciNet  Google Scholar 

  6. B.M. Levitan, Generalized Translation Operators and Some of Their Applications (Nauka, Moscow, 1962) (in Russian)

    MATH  Google Scholar 

  7. V.A. Kakichev, On the convolutions for integral transforms. Vescì Akad. Navuk BSSR, Ser. Fiz.-Mat. Navuk 22, 48–57 (1967) (in Russian)

    Google Scholar 

  8. R.V. Churchill, Fourier Series and Boundary Value Problems (McGraw-Hill, New York, 1941)

    Google Scholar 

  9. X.T. Nguyen, N.T. Hai, Convolution for Integral Transforms and Their Applications (Computer Center of the Russian Academy, Moscow, 1997)

    Google Scholar 

  10. X.T. Nguyen, V.A. Kakichev, V. Kim Tuan, On the generalized convolutions for Fourier cosine and sine transform. East-West J. Math. 1(1), 85–90 (1998)

    MATH  MathSciNet  Google Scholar 

  11. B.M. Levitan, Expansion in Fourier series and integrals with Bessel functions. Usp. Mat. Nauk 6:2(42), 102–143 (1951)

    MATH  MathSciNet  Google Scholar 

  12. Ya.I. Zhitomirskii, The Cauchy problem for systems of linear partial differential equations with differential operators of Bessel type. Math. USSR Sb. 36(2), 299–310 (1955)

    Google Scholar 

  13. I.I. Hirschman Jr., Variation diminishing Hankel transforms. J. Anal. Math. 8, 307–336 (1960/1961)

    Google Scholar 

  14. D.T. Haimo, Integral equations associated with Hankel convolutions. Trans. Am. Math. Soc. 116, 330–375 (1965)

    Article  MATH  MathSciNet  Google Scholar 

  15. F.M. Cholewinski, A Hankel convolution complex inversion theory. Mem. Am. Math. Soc. 58, 67 (1965)

    MathSciNet  Google Scholar 

  16. V.K. Tuan, M. Saigo, Convolutions of Hankel transform and its application to an integral involving Bessel function of first kind. Int. J. Math. Math. Sci. 18(2), 545–550 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  17. L.E. Britvina, On polyconvolutions generated by the Hankel transform. Math. Notes 76(1), 18–24 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  18. L.E. Britvina, A generalized convolution for Hankel transform with Gegenbauer’s polynomial in the kernel. Vestn. NovSU 30, 48–51 (2005) (in Russian)

    Google Scholar 

  19. L.E. Britvina, Generalized convolutions for the Hankel transform and related integral operators. Math. Nachr. 280(9-10), 962–970 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  20. L.E. Britvina, Integral operators related to generalized convolutions for Hankel transform. Integral Transforms Spec. Funct. 20(10), 785–796 (2009)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Novgorod State University, ul. B. St. Petersburgskaya, 41, 173003, Veliky Novgorod, Russia

    Lyubov Y. Britvina

Authors
  1. Lyubov Y. Britvina
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lyubov Y. Britvina .

Editor information

Editors and Affiliations

  1. Dept. of Computer Sc. & Comp. Methods, Pedagogical University, Kraków, Poland

    Vladimir V. Mityushev

  2. Imperial College London, London, United Kingdom

    Michael V. Ruzhansky

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this paper

Cite this paper

Britvina, L.Y. (2015). Generalized Shift Operators Generated by Convolutions of Integral Transforms. In: Mityushev, V., Ruzhansky, M. (eds) Current Trends in Analysis and Its Applications. Trends in Mathematics(). Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-12577-0_56

Download citation

Publish with us

Policies and ethics

Search

Navigation