The Hankel transform of tempered Boehmians via the exchange property. (English) Zbl 1302.46029
The authors investigate the Hankel transform of tempered Boehmians. A localization principle for distributions in \(\mathcal{B}_{\mu}^{'}\) is given in Section 2. The proofs of the main results are given in Section 3, whereas in Section 4, algebraic properties and convergence in \(\chi_{\mu}\) are dealt with.
Remarks: In Ref.[19] [A. Singh et al., Int.J. Math.Anal., Ruse 4, No. 45–48, 2199–2210 (2010; Zbl 1250.46028)], the convolution of an integral transform is described in Eqn.(7), and using the relation between the two-dimensional Fourier transformation and the Hankel transform, tempered Boehmians are defined. Two theorems are proved for tempered Boehmians, and as the definition of Boehmians relies on convolution, these theorems are proved based on Eqn.(7) using the Hankel transform of order zero. The use of convolution is mandatory to define Boehmians.
Remarks: In Ref.[19] [A. Singh et al., Int.J. Math.Anal., Ruse 4, No. 45–48, 2199–2210 (2010; Zbl 1250.46028)], the convolution of an integral transform is described in Eqn.(7), and using the relation between the two-dimensional Fourier transformation and the Hankel transform, tempered Boehmians are defined. Two theorems are proved for tempered Boehmians, and as the definition of Boehmians relies on convolution, these theorems are proved based on Eqn.(7) using the Hankel transform of order zero. The use of convolution is mandatory to define Boehmians.
Reviewer: Abhishek Singh (Jodhpur)
MSC:
46F12 | Integral transforms in distribution spaces |
44A35 | Convolution as an integral transform |
46F99 | Distributions, generalized functions, distribution spaces |
Citations:
Zbl 1250.46028References:
[1] | Atanasiu, D.; Mikusinski, P., On the Fourier transform and the exchange property, Int. J. Math. Math. Sci., 2005, 16, 2579-2584 (2005) · Zbl 1099.46028 |
[2] | D. Atanasiu, D. Nemzer, Extending the Laplace transform, Math. Student 77 (1-4) (2008/09) 203-212.; D. Atanasiu, D. Nemzer, Extending the Laplace transform, Math. Student 77 (1-4) (2008/09) 203-212. · Zbl 1196.44001 |
[3] | Betancor, J. J.; Linares, M.; Méndez, J. M., The Hankel transform of tempered Boehmians, Boll. Un. Mat. Ital. B (7), 10, 2, 325-340 (1996) · Zbl 0869.46018 |
[4] | Betancor, J. J.; Marrero, I., The Hankel convolution and the Zemanian spaces \(\beta_\mu\) and \(\beta_\mu^\prime \), Math. Nachr., 160, 277-298 (1993) · Zbl 0796.46023 |
[5] | Boehme, T. K., The support of Mikusinski operators, Trans. Am. Math. Soc., 176, 319-334 (1973) · Zbl 0268.44005 |
[6] | Erdelyi, A.; Magnus, W.; Oberhettinger, F.; Tricomi, F., Tables of Integral Transforms, vol. 2 (1954), McGraw-Hill · Zbl 0055.36401 |
[7] | Haimo, D. T., Integral equations associated with Hankel convolutions, Trans. Am. Math. Soc., 116, 330-375 (1965) · Zbl 0135.33502 |
[8] | Hirschman, I. I., Variation diminishing Hankel transforms, J. Anal. Math., 8, 307-336 (1960/61) · Zbl 0099.31301 |
[9] | Loonker, D.; Banerji, P. K.; Debnath, L., On the Hankel transform for Boehmians, Integr. Transf. Spec. Funct., 21, 7-8, 479-486 (2010) · Zbl 1207.44005 |
[10] | Kalpakam, N.; Ponnusamy, S., Convolution transform for Boehmians, Rocky Mountain J. Math., 33, 4, 1353-1378 (2003) · Zbl 1066.46032 |
[11] | Karunakaran, V.; Kalpakam, N., Hilbert transform for Boehmians, Integr. Transf. Spec. Funct., 9, 1, 19-36 (2000) · Zbl 0982.46030 |
[12] | Karunakaran, V.; Kalpakam, N., Weierstrass transform for Boehmians, Int. J. Math. Game Theory Algebra, 10, 3, 183-201 (2000) · Zbl 0963.46023 |
[13] | Marrero, I.; Betancor, J. J., Hankel convolution of generalized functions, Rend. Mat. Appl. (7), 15, 3, 351-380 (1995) · Zbl 0833.46026 |
[14] | Mikusinski, J.; Mikusinski, P., Quotients de suites et leurs applications dans l’analyse fonctionelle, C.R. Acad. Sci. Paris Ser. I Math., 293, 9, 463-464 (1981) · Zbl 0495.44006 |
[15] | Mikusinski, P., Tempered Boehmians and ultradistributions, Proc. Am. Math. Soc., 123, 3, 813-817 (1995) · Zbl 0821.46053 |
[16] | Nemzer, D., Integrable Boehmians, Fourier transforms and Poisson’s summation formula, Appl. Anal. Discrete Math., 1, 1, 172-183 (2007) · Zbl 1199.46096 |
[17] | Roopkumar, R., Mellin transform for Boehmians, Bull. Inst. Math. Acad. Sin. (N.S.), 4, 1, 75-96 (2009) · Zbl 1182.46030 |
[18] | Roopkumar, R.; Negrín, E., Poisson transform on Boehmians, Appl. Math. Comput., 216, 9, 2740-2748 (2010) · Zbl 1207.46037 |
[19] | Singh, A.; Loonker, D.; Banerji, P. K., Fourier-Bessel transform for tempered Boehmians, Int. J. Math. Anal. (Ruse), 4, 45-48, 2199-2210 (2010) · Zbl 1250.46028 |
[20] | de Sousa Pinto, J., A generalised Hankel convolution, SIAM J. Math. Anal., 16, 6, 1335-1346 (1985) · Zbl 0592.46038 |
[21] | Zemanian, A. H., The Hankel transformation of certain distributions of rapid growth, SIAM J. Appl. Math., 14, 678-690 (1966) · Zbl 0154.13804 |
[22] | Zemanian, A. H., Generalized Integral Transformations (1968), Interscience Publishers · Zbl 0181.12701 |
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