RETRACTED ARTICLE: Approximation of analytic functions by bessel functions of fractional order

This article was retracted on 01 July 2020

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We solve the inhomogeneous Bessel differential equation

$$ {x^2}y''(x) + xy'(x) + \left( {{x^2} - {\nu^2}} \right)y(x) = \sum\limits_{m = 0}^\infty {{a_m}{x^m},} $$

where ν is a positive nonintegral number, and use this result for the approximation of analytic functions of a special type by the Bessel functions of fractional order.

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  • 30 September 2020

    This article has been retracted. Please see the Retraction Notice for more detail: https://doi.org/10.1007/s11253-020-01784-z.

References

  1. 1.

    C. Alsina and R. Ger, “On some inequalities and stability results related to the exponential function,” J. Inequal. Appl., 2, 373–380 (1998).

    MathSciNet  MATH  Google Scholar 

  2. 2.

    T. Aoki, “On the stability of the linear transformation in Banach spaces,” J. Math. Soc. Jpn., 2, 64–66 (1950).

    MathSciNet  Article  Google Scholar 

  3. 3.

    S. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific, Singapore (2002).

    Book  Google Scholar 

  4. 4.

    G. L. Forti, “Hyers–Ulam stability of functional equations in several variables,” Aequat. Math., 50, 143–190 (1995).

    MathSciNet  Article  Google Scholar 

  5. 5.

    P. Găvrută, “A generalization of the Hyers–Ulam–Rassias stability of approximately additive mappings,” J. Math. Anal. Appl., 184, 431–436 (1994).

    MathSciNet  Article  Google Scholar 

  6. 6.

    D. H. Hyers, “On the stability of the linear functional equation,” Proc. Nat. Acad. Sci. USA, 27, 222–224 (1941).

    MathSciNet  Article  Google Scholar 

  7. 7.

    D. H. Hyers, G. Isac, and T. M. Rassias, Stability of Functional Equations in Several Variables, Birkhäuser, Boston (1998).

    Book  Google Scholar 

  8. 8.

    D. H. Hyers and T. M. Rassias, “Approximate homomorphisms,” Aequat. Math., 44, 125–153 (1992).

    MathSciNet  Article  Google Scholar 

  9. 9.

    S.-M. Jung, Hyers–Ulam–Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press, Palm Harbor (2001).

    MATH  Google Scholar 

  10. 10.

    S.-M. Jung, “Legendre’s differential equation and its Hyers–Ulam stability,” Abstr. Appl. Anal., 2007, Article ID 56419 (2007), doi:10.1155/2007/56419.

  11. 11.

    S.-M. Jung, “Approximation of analytic functions by Airy functions,” Integral Transforms Spec. Funct., 19, No. 12, 885–891 (2008).

    MathSciNet  Article  Google Scholar 

  12. 12.

    S.-M. Jung, “An approximation property of exponential functions,” Acta Math. Hungar., 124, No. 1–2, 155–163 (2009).

    MathSciNet  Article  Google Scholar 

  13. 13.

    S.-M. Jung, “Approximation of analytic functions by Hermite functions,” Bull. Sci. Math., 2009, 133, No. 7, 756–764.

    MathSciNet  Article  Google Scholar 

  14. 14.

    S.-M. Jung, “Approximation of analytic functions by Legendre functions,” Nonlin. Anal., 71, No. 12, 103–108 (2009).

    MathSciNet  Article  Google Scholar 

  15. 15.

    S.-M. Jung and S. Min, “On approximate Euler differential equations,” Abstr. Appl. Anal., 2009, Article ID 537963 (2009).

  16. 16.

    B. Kim and S.-M. Jung, “Bessel’s differential equation and its Hyers–Ulam stability,” J. Inequal. Appl., 2007, Article ID 21640 (2007).

  17. 17.

    E. Kreyszig, Advanced Engineering Mathematics, Wiley, New York (1979).

    MATH  Google Scholar 

  18. 18.

    S. Lang, Undergraduate Analysis, Springer, New York (1997).

    Book  Google Scholar 

  19. 19.

    T. Miura, S.-M. Jung, and S.-E. Takahasi, “Hyers–Ulam–Rassias stability of the Banach space valued linear differential equations y′ = λy;” J. Korean Math. Soc., 41, 995–1005 (2004).

    MathSciNet  Article  Google Scholar 

  20. 20.

    T. Miura, S. Miyajima, and S.-E. Takahasi, “Hyers–Ulam stability of linear differential operator with constant coefficients,” Math. Nachr., 258, 90–96 (2003).

    MathSciNet  Article  Google Scholar 

  21. 21.

    T. Miura, S. Miyajima, and S.-E. Takahasi, “A characterization of Hyers–Ulam stability of first order linear differential operators,” J. Math. Anal. Appl., 286, 136–146 (2003).

    MathSciNet  Article  Google Scholar 

  22. 22.

    M. Obłoza, “Hyers stability of the linear differential equation,” Rocz. Nauk.-Dydakt. Prace Mat., 13, 259–270 (1993).

    MathSciNet  MATH  Google Scholar 

  23. 23.

    M. Obłoza, “Connections between Hyers and Lyapunov stability of the ordinary differential equations,” Rocz. Nauk.-Dydakt. Prace Mat., 14, 141–146 (1997).

    MathSciNet  MATH  Google Scholar 

  24. 24.

    T. M. Rassias, “On the stability of the linear mapping in Banach spaces,” Proc. Am. Math. Soc., 72, 297–300 (1978).

    MathSciNet  Article  Google Scholar 

  25. 25.

    T. M. Rassias, “On the stability of functional equations and a problem of Ulam,” Acta Appl. Math., 62, 23–130 (2000).

    MathSciNet  Article  Google Scholar 

  26. 26.

    S.-E. Takahasi, T. Miura, and S. Miyajima, “On the Hyers–Ulam stability of the Banach space-valued differential equation y′ = λy;” Bull. Korean Math. Soc., 39, 309–315 (2002).

    MathSciNet  Article  Google Scholar 

  27. 27.

    S. M. Ulam, A Collection of Mathematical Problems, Interscience, New York (1960).

    MATH  Google Scholar 

  28. 28.

    W. R. Wade, An Introduction to Analysis, Prentice Hall, Upper Saddle River, NJ (2000).

    MATH  Google Scholar 

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Correspondence to S.-M. Jung.

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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 63, No. 12, pp. 1699–1709, December, 2011.

This article has been retracted. Please see the retraction notice for more detail:https://doi.org/10.1007/s11253-020-01784-z"

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Jung, SM. RETRACTED ARTICLE: Approximation of analytic functions by bessel functions of fractional order. Ukr Math J 63, 1933–1944 (2012). https://doi.org/10.1007/s11253-012-0622-4

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Keywords

  • Functional Equation
  • Bessel Function
  • Fractional Order
  • Differential Inequality
  • Linear Differential Operator