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Hankel-Schwartz and Hankel-Clifford transforms on \({\cal L}_{\nu,p}\)-spaces

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Abstract

The paper is devoted to study Map** properties of the Hankel-Schwartz integral transform and the Hankel-Clifford integral transform

$$ ({\rm H}_{\eta,1}^{s}f)(x)=x^{-\eta}\int_0^\infty J_{\eta}(xt)t^{n+1}f(t)dt(\eta \in {\rm C,Re}(\eta)>-1,x>0) $$
$$ ({\rm H}_{\eta,1}^{s}f)(x)=x^{\eta /2}\int_0^\infty J_{\eta}\bigg (2(xt)1/2\bigg )t^{n/2}f(t)dt(\eta \in {\rm C,Re}(\eta)>-1,x>0) $$

on certain spaces \({\cal L}_{\nu,p}(\nu \in {\rm R}=(-\infty,\infty );1\leq p\leq \infty )\) of measurable functions. Their representations in the form of integral transforms with H-function kernels are presented.

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References

  1. Erdelyi A., Magnus W., Oberhettinger F. and Tricomi F. G. Higher Transcendental functions, Vol.1, McGraw-Hill, Neqw York-Toronto -London, 1953.

    Google Scholar 

  2. Hayek, N., A study of the differential equation \( xy {\prime \prime }+(\nu+1) {\prime }+y=0 \) and its applications. (Spanish) Collect Math. 18 (1966-67), no. 1–2, 57–174.

    MathSciNet  Google Scholar 

  3. Schwartz A. L., An inversion theorem for Hankel transforms, Proc. Amer. Math. Soc. 22 (1969), no. 3, 713–717.

    Article  MathSciNet  MATH  Google Scholar 

  4. Lee W. Y., On Schwartz’s Hankel transformation of certain spaces of distributions, SIAM J. Math. Anal 6 (1975), no.2, 427–432.

    Article  MathSciNet  MATH  Google Scholar 

  5. Dube L. S. and Pandey J. N., On the Hankel transform of distributions, Tohôku Math. J. (2) 27 (1975), no.3, 337–354.

    Article  MathSciNet  MATH  Google Scholar 

  6. Altenburg G., Bessel-Transformationen in Räumen von Grundfunktionen über dem Intervall Ω = (0,∞) und deren Daulräumen, Math. Nachr. 108 (1982), 197–218.

    Article  MathSciNet  MATH  Google Scholar 

  7. Mendez J. M., On the Besel transformation of arbitrary order, Math. Nachr. 136 (1988), 233–239.

    Article  MathSciNet  MATH  Google Scholar 

  8. Mendez J. M., A mixed Parseval equation and the generalized Bessel transformation, Proc. Amer. Math. Soc. 102 (1988), no. 3, 619–624.

    Article  MathSciNet  MATH  Google Scholar 

  9. Mendez J. M. and Sanchez Quintana A. M., On the Bessel transformation of rapidly increasing generalized functions, Ranchi Univ. Math. J. 19 (1988), 13–24.

    MathSciNet  MATH  Google Scholar 

  10. Mendez Perez J. M. R. and Sanchez Quintana A. M., On the Schwartz’s Hankel transformation of distributions, Analysis 13 (1993), no.1–2, 1–18.

    MathSciNet  MATH  Google Scholar 

  11. Sanchez Quintana A. M. and Mendez Perez J. M. R., The Schwartz’s Hankel transformations of disributions of rapid growth and a mixed Parceval equation, Bull. Soc. Roy. Sci. Liege 59 (1990), no. 2, 183–204.

    MathSciNet  MATH  Google Scholar 

  12. Betancor J. J. and Negrin E. R., The generalized Bessel transformations on the spaces \( L_{p,\nu} {\prime} \) of distributions, J. Korean Math. Soc. 27 (1990), no.1, 129–135.

    MathSciNet  MATH  Google Scholar 

  13. Betancor J. J., On the Hankel-Schwartz integral transform (Spanish), Rev. Colombiana Mat. 24 (1990), no.1-2, 15–23.

    MathSciNet  MATH  Google Scholar 

  14. Linares L. M. and Mendez J. M., Hankel complementary integral transformations of arbitrary order, Internat. J. Math. Math. Sci. 15 (1992), no.2, 323–332.

    Google Scholar 

  15. Brychkov Ya.A. and Prudnikov A. P., Integral Transforms of Generalized Functions, Gordon and Breach, Yverdon et alibi, 1989.

  16. Betancor J. J., On Hankel-Clifford transforms of ultradistributions, Pure Appl. Math. Sci. 29 (1989), no.1-2, 21–43.

    MathSciNet  MATH  Google Scholar 

  17. Betancor J. J., Two complex variants of a Hankel type transformation of generalized functions, Portugal. Math. 46 (1989), no.3, 229–243.

    MathSciNet  MATH  Google Scholar 

  18. Mendez Perez J. M. R. and Sosas Robayna M. M., A pair of generalized Hankel-Clifford transformations and their applications, J. Math. Anal. Appl. 154 (1991), no.2, 543–557.

    Article  MathSciNet  MATH  Google Scholar 

  19. Kilbas A. A and Trujilllo J. J., Generalized Hankel transform on \({\cal L}_{\nu,r}\)-spaces, Integral Transform. Spec. Funct. 9 (2000), no. 1, 67–85.

    Google Scholar 

  20. Rooney P. G., A technique for studying the boundednes and extendability of certain types of operatotrs, Can. J. Math. 25 (1973), no.5, 1090–1102.

    Article  MathSciNet  MATH  Google Scholar 

  21. Rooney P. G., On the range of the Hankel transform, Bull. London Math. Soc. 11 (1979), 45–48.

    Article  MathSciNet  MATH  Google Scholar 

  22. Rooney P. G., On the \({\cal Y}_{\nu}\) and \({\cal H}_{\nu}\) transfromations, Can. J. Math. 32 (1980), no.5, 1021–1044.

    Article  MathSciNet  MATH  Google Scholar 

  23. Rooney P. G., On the representation of functions by the Hankel and some related transformations, Proc. Royal Soc. Edinburgh 125A (1995), 449–463.

    Article  MathSciNet  Google Scholar 

  24. Heywood P. and Rooney P. G., On the Hankel and some related transformations, Can. J. Math. 40 (1988), no.4, 989–1009.

    Article  MathSciNet  MATH  Google Scholar 

  25. Mathai A. M. and Saxena R. K., The H-Function with Applications in Statistics and Other Disciplines, Haldsted Press [John Wiley and Sons], New York-London-Sydney, 1978.

    MATH  Google Scholar 

  26. Prudnikov A. P., Brychkov Yu.A. and Marichev O. I., Integrals and Series, Vol.3, More Special Functions, Gordon and Breach, New York et alibi, 1990.

    Google Scholar 

  27. Srivastava H. M., Gupta K. C. and Goyal S. L., The H-Function of One and Two Variables with Applications, South Asian Publishers, New Delhi-Madras, 1982.

    Google Scholar 

  28. Kilbas A. A., Saigo M. and Borovco A. N., On the generalized Hardy-Titchmarsh transform in \({\cal L}_{\nu,r}\)-space, Fukuoka Univ. Sci. Rep. 30 (2000), no.l, 67–85.

    MathSciNet  MATH  Google Scholar 

  29. Kilbas A. A., Saigo M. and Borovco A. N., The generalized Hardy-Titchmarsh transform in the space of summable functions. (Russian) Dokl. Acad. Nauk 372 (2000), no. 4, 451–454.

    Google Scholar 

  30. Kilbas A. A., Saigo M. and Shlapakov S. A., Integral transforms with Fox+s H-function in spaces of summable functions, Integral Transform. Spec. Fund. 1 (1993), no.2, 87–103.

    Article  MathSciNet  MATH  Google Scholar 

  31. Kilbas A. A., Saigo M. and Shlapakov S. A., Integral transforms with Foxs H-function in \({\cal L}_{\nu,p}\)-spaces, Fukuoka Univ. Sci. Rep. 23 (1993), no.l, 9–31.

    MathSciNet  MATH  Google Scholar 

  32. Kilbas A. A., Saigo M. and Shlapakov S. A., Integral transforms with Foxs H -function in \({\cal L}_{\nu,p}\)-spaces, II Fukuoka Univ. Sci. Rep. 24 (1994), no.l, 13–38.

    MathSciNet  MATH  Google Scholar 

  33. Glaeske H. -J., Kilbas A. A., Saigo M. and Shlapakov S. A., Integral transforms with H-function kernels on \({\cal L}_{\nu,r}\)-spaces, Appl. Anal, to appear.

  34. Betancor J. J. and Jerez Diaz G, Boundedness and range of \({\cal H}\)-transformation on certain weighted Lp-spaces, Serdica 20 (1994), no. 3-4, 269–297.

    MathSciNet  MATH  Google Scholar 

  35. Shlapakov S. A., Saigo M. and Kilbas A. A., On inversion of H-transform in \({\cal L}_{\nu,r}\) -space, Intern. J. Math. Math. Sci. 21 (1998), no.4, 713–722.

    Article  MathSciNet  MATH  Google Scholar 

  36. Rooney P. G., On integral transformations with G-function kernels, Proc. Royal Soc. Edinburgh, Sect. A 93 (1982/83), 265–297.

    MathSciNet  Google Scholar 

  37. Erdelyi A., Magnus W., Oberhettinger F. and Tricomi F. G. Higher Transcendental functions, Vol.11, McGraw-Hill, Neqw York-Toronto -London, 1953.

    Google Scholar 

  38. Samko S. G., Kilbas A. A. and Marichev O. I., Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach, Yverdon et alibi, 1993.

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Authors and Affiliations

  1. Department of Mathematics and Mechanics, Belarusian State University, 220050, Minsk, Belarus

    Anatoly A. Kilbas

  2. Departamento de Análisis Matemático, Universidad de La Laguna, 38271, La Laguna-Tenerife, Spain

    Juan J. Trujillo

Authors
  1. Anatoly A. Kilbas
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  2. Juan J. Trujillo
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Kilbas, A.A., Trujillo, J.J. Hankel-Schwartz and Hankel-Clifford transforms on \({\cal L}_{\nu,p}\)-spaces. Results. Math. 43, 284–299 (2003). https://doi.org/10.1007/BF03322743

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