| Title:
             | 
On Hankel transform and Hankel convolution of Beurling type distributions having upper bounded support (English) | 
| Author:
             | 
Belhadj, M. | 
| Author:
             | 
Betancor, J. J. | 
| Language:
             | 
English | 
| Journal:
             | 
Czechoslovak Mathematical Journal | 
| ISSN:
             | 
0011-4642 (print) | 
| ISSN:
             | 
1572-9141 (online) | 
| Volume:
             | 
54 | 
| Issue:
             | 
2 | 
| Year:
             | 
2004 | 
| Pages:
             | 
315-336 | 
| Summary lang:
             | 
English | 
| . | 
| Category:
             | 
math | 
| . | 
| Summary:
             | 
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  ${\mathcal E}(w)^{\prime }$. (English) | 
| Keyword:
             | 
Beurling distributions | 
| Keyword:
             | 
Hankel transformation | 
| Keyword:
             | 
convolution | 
| MSC:
             | 
44A15 | 
| MSC:
             | 
46F10 | 
| MSC:
             | 
46F12 | 
| idZBL:
             | 
Zbl 1080.46025 | 
| idMR:
             | 
MR2059253 | 
| . | 
| Date available:
             | 
2009-09-24T11:12:52Z | 
| Last updated:
             | 
2020-07-03 | 
| Stable URL:
             | 
http://hdl.handle.net/10338.dmlcz/127890 | 
| . | 
| Reference:
             | 
[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 0695127 | 
| Reference:
             | 
[2] M. Belhadj and J. J. Betancor: Beurling distributions and Hankel transforms.Math. Nachr 233-234 (2002), 19–45. MR 1879861 | 
| Reference:
             | 
[3] M. Belhadj and J. J. Betancor: Hankel transformation and Hankel convolution of tempered Beurling distributions.Rocky Mountain J.  Math 31 (2001), 1171–1203. MR 1895292, 10.1216/rmjm/1021249437 | 
| Reference:
             | 
[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 1245003 | 
| Reference:
             | 
[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 1250341 | 
| Reference:
             | 
[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. MR 1619757, 10.1080/10652469508819075 | 
| Reference:
             | 
[7] J. J. Betancor and L. Rodríguez-Mesa: Hankel convolution on distribution spaces with exponential growth.Studia Math. 121 (1996), 35–52. MR 1414893, 10.4064/sm-121-1-35-52 | 
| Reference:
             | 
[8] A. Beurling: Quasi-analyticity and General Distributions. Lectures  4 and 5.A.M.S. Summer Institute, Stanford, 1961. | 
| Reference:
             | 
[9] G. Björck: Linear partial differential operators and generalized distributions.Ark. Math. 6 (1966), 351–407. MR 0203201, 10.1007/BF02590963 | 
| Reference:
             | 
[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 1762416 | 
| Reference:
             | 
[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. MR 1059516, 10.1007/BF01199116 | 
| Reference:
             | 
[12] R. W. Braun, R. Meise and B. A. Taylor: Ultradifferentiable functions and Fourier analysis.Results in Maths. 17 (1990), 206–237. MR 1052587, 10.1007/BF03322459 | 
| Reference:
             | 
[13] F. M. Cholewinski: A Hankel convolution complex inversion theory.Mem. Amer. Math. Soc. 58 (1965). Zbl 0137.30901, MR 0180813 | 
| Reference:
             | 
[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. | 
| Reference:
             | 
[15] D. T. Haimo: Integral equations associated with Hankel convolutions.Trans. Amer. Math. Soc. 116 (1965), 330–375. Zbl 0135.33502, MR 0185379, 10.1090/S0002-9947-1965-0185379-4 | 
| Reference:
             | 
[16] C. S. Herz: On the mean inversion of Fourier and Hankel transforms.Proc. Nat. Acad. Sci. USA, 40 (1954), 996–999. Zbl 0059.09901, MR 0063477 | 
| Reference:
             | 
[17] I. I. Hirschman,  Jr.: Variation diminishing Hankel transforms.J.  Analyse Math. 8 (1960/61), 307–336. MR 0157197 | 
| Reference:
             | 
[18] L. Hörmander: Hypoelliptic convolution equations.Math. Scand. 9 (1961), 178–184. MR 0139838, 10.7146/math.scand.a-10633 | 
| Reference:
             | 
[19] I. Marrero and J. J. Betancor: Hankel convolution of generalized functions.Rendiconti di Matematica 15 (1995), 351–380. MR 1362778 | 
| Reference:
             | 
[20] J. M. Méndez: On the Bessel transforms of arbitrary order.Math. Nachr. 136 (1988), 233–239. MR 0952475, 10.1002/mana.19881360116 | 
| Reference:
             | 
[21] J. M. Méndez and A. M. Sánchez: On the Schwartz’s Hankel transformation of distributions.Analysis 13 (1993), 1–18. 10.1524/anly.1993.13.12.1 | 
| Reference:
             | 
[22] L. Schwartz: Theorie des distributions.Hermann, Paris, 1978. Zbl 0399.46028, MR 0209834 | 
| Reference:
             | 
[23] J. de Sousa-Pinto: A generalized Hankel convolution.SIAM J.  Appl. Math. 16 (1985), 1335–1346. Zbl 0592.46038, MR 0807914, 10.1137/0516097 | 
| Reference:
             | 
[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. Zbl 0591.42014, MR 0849618 | 
| Reference:
             | 
[25] G. N. Watson: A Treatise on the Theory of Bessel Functions.Cambridge University Press, Cambridge, 1959. MR 1349110 | 
| Reference:
             | 
[26] A. H. Zemanian: A distributional Hankel transformation.SIAM J.  Appl. Math. 14 (1966), 561–576. Zbl 0154.13803, MR 0201930, 10.1137/0114049 | 
| Reference:
             | 
[27] A. H. Zemanian: The Hankel transformation of certain distribution of rapid growth.SIAM J.  Appl. Math. 14 (1966), 678–690. MR 0211211, 10.1137/0114056 | 
| Reference:
             | 
[28] A. H. Zemanian: Generalized Integral Transformations.Interscience Publishers, New York, 1968. Zbl 0181.12701, MR 0423007 | 
| . |