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New spaces of functions and hyperfunctions for Hankel transforms and convolutions

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Abstract.

In this paper we study the Hankel transformation and convolution on certain spaces \({\cal G}_{e}\) of entire functions and its dual \({\cal G}_{e}{\prime}\) that is a space of hyperfunctions and contains the (even)-Schwartz space S e ′. We prove that the Hankel transform is an automorphism of \({\cal G}_{e}{\prime}\). Also the Hankel convolutors of \({\cal G}_{e}{\prime}\) are investigated.

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References

  • G Altenburg (1982) ArticleTitleBessel-Transformationen in Räumen von Grundfunktionen über dem Intervall Ω = (0, ∞) und deren Dualräumen Math Nachr 108 197–218 Occurrence Handle0528.46035 Occurrence Handle10.1002/mana.19821080116 Occurrence Handle695127

    Article  MATH  MathSciNet  Google Scholar 

  • M Belhadj JJ Betancor (2002) ArticleTitleBeurling distributions and Hankel transforms Math Nachr 233–234 19–45 Occurrence Handle10.1002/1522-2616(200201)233:1<19::AID-MANA19>3.0.CO;2-B Occurrence Handle1879861

    Article  MathSciNet  Google Scholar 

  • JJ Betancor B González (1995) ArticleTitleA convolution operation for a distributional Hankel transformation Stud Math 117 57–72 Occurrence Handle0837.46029

    MATH  Google Scholar 

  • JJ Betancor C Jerez S Molina L Rodríguez-Mesa (2007) ArticleTitleDistributional convolutors for Fourier transform J Math Anal Appl 325 459–468 Occurrence Handle1126.46028 Occurrence Handle10.1016/j.jmaa.2006.01.066 Occurrence Handle2273537

    Article  MATH  MathSciNet  Google Scholar 

  • JJ Betancor L Rodríguez-Mesa (1996) ArticleTitleHankel convolution on distribution spaces with exponential growth Stud Math 121 35–52 Occurrence Handle0862.46021

    MATH  Google Scholar 

  • von Eijndhoven SJL, Kerkhof MJ (1988) The Hankel transformation and spaces of type W. Reports on Appl and Num Analysis 10, Dept of Maths and Comp Sci. Eindhoven: Univ Technology

  • DT Haimo (1965) ArticleTitleIntegral equations associated with Hankel convolutions Trans Amer Math Soc 116 330–375 Occurrence Handle0135.33502 Occurrence Handle10.2307/1994121 Occurrence Handle185379

    Article  MATH  MathSciNet  Google Scholar 

  • C Herz (1954) ArticleTitleOn the mean inversion of Fourier and Hankel transforms Proc Natl Acad Sci USA 40 996–999 Occurrence Handle0059.09901 Occurrence Handle10.1073/pnas.40.10.996 Occurrence Handle63477

    Article  MATH  MathSciNet  Google Scholar 

  • II Hirschman SuffixJr (1961) ArticleTitleVariation diminishing Hankel transforms J Anal Math 8 307–336 Occurrence Handle0099.31301 Occurrence Handle157197 Occurrence Handle10.1007/BF02786854

    Article  MATH  MathSciNet  Google Scholar 

  • KB Howell (1992) ArticleTitleA new theory for Fourier analysis. I: The space of test functions J Math Anal Appl 168 342–350 Occurrence Handle0776.42012 Occurrence Handle10.1016/0022-247X(92)90162-7 Occurrence Handle1175993

    Article  MATH  MathSciNet  Google Scholar 

  • KB Howell (1993) ArticleTitleA new theory for Fourier analysis. II: Further analysis on the space of test functions J Math Anal Appl 173 419–429 Occurrence Handle0776.42013 Occurrence Handle10.1006/jmaa.1993.1076 Occurrence Handle1209328

    Article  MATH  MathSciNet  Google Scholar 

  • KB Howell (1993) ArticleTitleA new theory for Fourier analysis. III: Basic analysis on the dual J Math Anal Appl 175 257–267 Occurrence Handle0776.42014 Occurrence Handle10.1006/jmaa.1993.1166 Occurrence Handle1216759

    Article  MATH  MathSciNet  Google Scholar 

  • KB Howell (1993) ArticleTitleA new theory for Fourier analysis. IV: Basic multiplication and convolution on the dual space J Math Anal Appl 180 79–92 Occurrence Handle0811.42004 Occurrence Handle10.1006/jmaa.1993.1384 Occurrence Handle1250279

    Article  MATH  MathSciNet  Google Scholar 

  • KB Howell (1994) ArticleTitleA new theory for Fourier analysis. V: Generalized multiplication and convolution on the dual space J Math Anal Appl 187 567–582 Occurrence Handle0819.46028 Occurrence Handle10.1006/jmaa.1994.1375 Occurrence Handle1297043

    Article  MATH  MathSciNet  Google Scholar 

  • Y Ito (1979) ArticleTitleOn the theory of vector valued hyperfunctions J Math Tokushima Univ 13 29–51 Occurrence Handle0429.46027 Occurrence Handle563394

    MATH  MathSciNet  Google Scholar 

  • Y Ito (1988) ArticleTitleFourier hyperfunctions of general type J Math Kyoto Univ 28 213–265 Occurrence Handle0711.46035 Occurrence Handle953176

    MATH  MathSciNet  Google Scholar 

  • A Kaneko (1988) Introduction to Hyperfunctions KTK Scientific Publ Tokyo Occurrence Handle0687.46027

    MATH  Google Scholar 

  • V Karunakaran T Venugopal (2000) ArticleTitleA new space for Fourier transform Integral Transforms Spec Funct 9 133–148 Occurrence Handle0967.46028 Occurrence Handle10.1080/10652460008819250 Occurrence Handle1784496

    Article  MATH  MathSciNet  Google Scholar 

  • T Kawai (1970) ArticleTitleOn the theory of Fourier hyperfunctions and its applications to partial differential equations with constant coefficients J Fac Sci Univ Tokyo, Sect IA Math 17 467–515 Occurrence Handle0212.46101 Occurrence Handle298200

    MATH  MathSciNet  Google Scholar 

  • KW Kim (2004) ArticleTitleDenseness of test function spaces of extended Fourier hyperfunctions Bull Korean Math Soc 41 785–803 Occurrence Handle1077.46039 Occurrence Handle2103125 Occurrence Handle10.4134/BKMS.2004.41.4.785

    Article  MATH  MathSciNet  Google Scholar 

  • KW Kim (2004) ArticleTitleThe space of Fourier hyperfunctions as an inductive limit of Hilbert spaces Commun Korean Math Soc 19 661–681 Occurrence Handle1101.46312 Occurrence Handle2096589

    MATH  MathSciNet  Google Scholar 

  • KW Kim (2005) ArticleTitleConvolutors for the space of Fourier hyperfunctions J Korean Math Soc 42 599–619 Occurrence Handle1085.46029 Occurrence Handle2134719 Occurrence Handle10.4134/JKMS.2005.42.3.599

    Article  MATH  MathSciNet  Google Scholar 

  • I Marrero JJ Betancor (1995) ArticleTitleHankel convolution of generalized functions Rend Mat Appl, VII Ser 15 351–380 Occurrence Handle0833.46026 Occurrence Handle1362778

    MATH  MathSciNet  Google Scholar 

  • Y Saburi (1985) ArticleTitleFundamental properties of modified Fourier hyperfunctions Tokyo J Math 8 231–273 Occurrence Handle0597.46038 Occurrence Handle800088 Occurrence Handle10.3836/tjm/1270151582

    Article  MATH  MathSciNet  Google Scholar 

  • M Sato (1959) ArticleTitleTheory of hyperfunctions, I J Fac Sci Univ Tokyo, Sec I 8 139–193 Occurrence Handle0087.31402

    MATH  Google Scholar 

  • M Sato (1960) ArticleTitleTheory of hyperfunctions, II J Fac Sci Univ Tokyo, Sec I 8 387–437 Occurrence Handle0097.31404

    MATH  Google Scholar 

  • GN Watson (1958) Theory of Bessel Functions Univ Press Cambridge

    Google Scholar 

  • AH Zemanian (1966) ArticleTitleA distributional Hankel transformation SIAM J Appl Math 14 561–576 Occurrence Handle0154.13803 Occurrence Handle10.1137/0114049 Occurrence Handle201930

    Article  MATH  MathSciNet  Google Scholar 

  • AH Zemanian (1987) Generalized integral transformations Dover New York Occurrence Handle0643.46029

    MATH  Google Scholar 

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Authors and Affiliations

  1. Universidad de la Laguna, La Laguna, España

    Jorge J. Betancor, Claudio Jerez & Lourdes Rodríguez-Mesa

  2. Universidad Nacional de Mar del Plata, Mar del Plata, Argentina

    Sandra M. Molina

Authors
  1. Jorge J. Betancor
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  2. Claudio Jerez
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  3. Sandra M. Molina
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  4. Lourdes Rodríguez-Mesa
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Additional information

Authors’ addresses: Jorge J. Betancor, Claudio Jerez and Lourdes Rodríguez-Mesa, Departamento de Análisis Matemático, Universidad de la Laguna, Campus de Anchieta, Avda. Astrofísico Francisco Sánchez, s/n, 38271 La Laguna (Sta. Cruz de Tenerife), España; Sandra M. Molina, Departamento de Matemáticas, Facultad de Ciencias Exactas y Naturales, Universidad Nacional de Mar del Plata, Funes 3350 (7600), Mar del Plata, Argentina

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Betancor, J., Jerez, C., Molina, S. et al. New spaces of functions and hyperfunctions for Hankel transforms and convolutions. Monatsh Math 153, 89–103 (2008). https://doi.org/10.1007/s00605-007-0498-9

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  • DOI: https://doi.org/10.1007/s00605-007-0498-9

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