Geodesic
On Hankel transform and Hankel convolution of Beurling type distributions having upper bounded support
Czechoslovak Mathematical Journal, Tome 54 (2004) no. 2, pp. 315-336

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MR Zbl
In this paper we study Beurling type distributions in the Hankel setting. We consider the space ${\mathcal E}(w)^{\prime }$ of Beurling type distributions on $(0, \infty )$ having upper bounded support. The Hankel transform and the Hankel convolution are studied on the space ${\mathcal E}(w)^{\prime }$. We also establish Paley Wiener type theorems for Hankel transformations of distributions in ${\mathcal E}(w)^{\prime }$.
In this paper we study Beurling type distributions in the Hankel setting. We consider the space ${\mathcal E}(w)^{\prime }$ of Beurling type distributions on $(0, \infty )$ having upper bounded support. The Hankel transform and the Hankel convolution are studied on the space ${\mathcal E}(w)^{\prime }$. We also establish Paley Wiener type theorems for Hankel transformations of distributions in ${\mathcal E}(w)^{\prime }$.
Classification : 44A15, 46F10, 46F12
Keywords: Beurling distributions; Hankel transformation; convolution 
Belhadj, M.; Betancor, J. J. On Hankel transform and Hankel convolution of Beurling type distributions having upper bounded support. Czechoslovak Mathematical Journal, Tome 54 (2004) no. 2, pp. 315-336. http://geodesic.mathdoc.fr/item/CMJ_2004_54_2_a4/
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     title = {On {Hankel} transform and {Hankel} convolution of {Beurling} type distributions having upper bounded support},
     journal = {Czechoslovak Mathematical Journal},
     pages = {315--336},
     year = {2004},
     volume = {54},
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     mrnumber = {2059253},
     zbl = {1080.46025},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2004_54_2_a4/}
}
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[1] G. Altenburg: Bessel transformationen in Raumen von Grundfunktionen uber dem Intervall $\Omega = (0, \infty )$ un derem Dualraumen. Math. Nachr. 108 (1982), 197–218. | MR

[2] M. Belhadj and J. J. Betancor: Beurling distributions and Hankel transforms. Math. Nachr 233-234 (2002), 19–45. | MR

[3] M. Belhadj and J. J. Betancor: Hankel transformation and Hankel convolution of tempered Beurling distributions. Rocky Mountain J.  Math 31 (2001), 1171–1203. | DOI | MR

[4] J. J. Betancor and I. Marrero: The Hankel convolution and the Zemanian spaces  $B_\mu $ and $B_\mu ^{\prime }$. Math. Nachr. 160 (1993), 277–298. | MR

[5] J. J. Betancor and I. Marrero: Structure and convergence in certain spaces of distributions and the generalized Hankel convolution. Math. Japon. 38 (1993), 1141–1155. | MR

[6] J. J. Betancor and I. Marrero: New spaces of type  $H_\mu $ and the Hankel transformation. Integral Transforms and Special Functions 3 (1995), 175–200. | DOI | MR

[7] J. J. Betancor and L. Rodríguez-Mesa: Hankel convolution on distribution spaces with exponential growth. Studia Math. 121 (1996), 35–52. | DOI | MR

[8] A. Beurling: Quasi-analyticity and General Distributions. Lectures  4 and 5. A.M.S. Summer Institute, Stanford, 1961.

[9] G. Björck: Linear partial differential operators and generalized distributions. Ark. Math. 6 (1966), 351–407. | DOI | MR

[10] J. Bonet, C. Fernández and R. Meise: Characterization of the $w$-hypoelliptic convolution operators on ultradistributions. Ann. Acad. Sci. Fenn. Mathematica 25 (2000), 261–284. | MR

[11] R. W. Braun and R. Meise: Generalized Fourier expansions for zero-solutions of surjective convolution operators in ${\mathcal D}_{\lbrace w\rbrace }(R)^{\prime }$. Arch. Math. 55 (1990), 55–63. | DOI | MR

[12] R. W. Braun, R. Meise and B. A. Taylor: Ultradifferentiable functions and Fourier analysis. Results in Maths. 17 (1990), 206–237. | DOI | MR

[13] F. M. Cholewinski: A Hankel convolution complex inversion theory. Mem. Amer. Math. Soc. 58 (1965). | MR | Zbl

[14] S. J. L. van Eijndhoven and M. J. Kerkhof: The Hankel transformation and spaces of type  $W$. Reports on Appl. and Numer. Analysis, 10. Dept. of Maths. and Comp. Sci., Eindhoven University of Technology, 1988.

[15] D. T. Haimo: Integral equations associated with Hankel convolutions. Trans. Amer. Math. Soc. 116 (1965), 330–375. | DOI | MR | Zbl

[16] C. S. Herz: On the mean inversion of Fourier and Hankel transforms. Proc. Nat. Acad. Sci. USA, 40 (1954), 996–999. | MR | Zbl

[17] I. I. Hirschman,  Jr.: Variation diminishing Hankel transforms. J.  Analyse Math. 8 (1960/61), 307–336. | MR

[18] L. Hörmander: Hypoelliptic convolution equations. Math. Scand. 9 (1961), 178–184. | DOI | MR

[19] I. Marrero and J. J. Betancor: Hankel convolution of generalized functions. Rendiconti di Matematica 15 (1995), 351–380. | MR

[20] J. M. Méndez: On the Bessel transforms of arbitrary order. Math. Nachr. 136 (1988), 233–239. | DOI | MR

[21] J. M. Méndez and A. M. Sánchez: On the Schwartz’s Hankel transformation of distributions. Analysis 13 (1993), 1–18. | DOI

[22] L. Schwartz: Theorie des distributions. Hermann, Paris, 1978. | MR | Zbl

[23] J. de Sousa-Pinto: A generalized Hankel convolution. SIAM J.  Appl. Math. 16 (1985), 1335–1346. | DOI | MR | Zbl

[24] K. Stempak: La theorie de Littlewood-Paley pour la transformation de Fourier-Bessel. C.R.  Acad. Sci. Paris 303 (Serie  I) (1986), 15–19. | MR | Zbl

[25] G. N. Watson: A Treatise on the Theory of Bessel Functions. Cambridge University Press, Cambridge, 1959. | MR

[26] A. H. Zemanian: A distributional Hankel transformation. SIAM J.  Appl. Math. 14 (1966), 561–576. | DOI | MR | Zbl

[27] A. H. Zemanian: The Hankel transformation of certain distribution of rapid growth. SIAM J.  Appl. Math. 14 (1966), 678–690. | DOI | MR

[28] A. H. Zemanian: Generalized Integral Transformations. Interscience Publishers, New York, 1968. | MR | Zbl