Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

ONE-PARAMETER GROUPS OF BOEHMIANS

  • Nemzer, Dennis (DEPARTMENT OF MATHEMATICS CALIFORNIA STATE UNIVERSITY)
  • Published : 2007.08.31
Download PDF
⟨ Previous Next ⟩

Abstract

The space of periodic Boehmians with $\Delta$-convergence is a complete topological algebra which is not locally convex. A family of Boehmians $\{T_\lambda\}$ such that $T_0$ is the identity and $T_{\lambda_1+\lambda_2}=T_\lambda_1*T_\lambda_2$ for all real numbers $\lambda_1$ and $\lambda_2$ is called a one-parameter group of Boehmians. We show that if $\{T_\lambda\}$ is strongly continuous at zero, then $\{T_\lambda\}$ has an exponential representation. We also obtain some results concerning the infinitesimal generator for $\{T_\lambda\}$.

Keywords

References

  1. P. K. Banerji and D. Loonker, On the Mellin transform of tempered Boehmians, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 62 (2000), no. 4, 39-48
  2. J. J. Betancor, M. Linares, and J. M. R. Mendez, The Hankel transform of integrable Boehmians, Appl. Anal. 58 (1995), no. 3-4, 367-382 https://doi.org/10.1080/00036819508840383
  3. J. J. Betancor, M. Linares, and J. M. R. Mendez, Ultraspherical transform of summable Boehmians, Math. Japon. 44 (1996), no. 1, 81-88
  4. J. Burzyk, P. Mikusinski, and D. Nemzer, Remarks on topological properties of Boehmi- ans, Rocky Mountain J. Math. 35 (2005), no. 3, 727-740 https://doi.org/10.1216/rmjm/1181069705
  5. N. V. Kalpakam and S. Ponnusamy, Convolution transform for Boehmians, Rocky Mountain J. Math. 33 (2003), no. 4, 1353-1378 https://doi.org/10.1216/rmjm/1181075468
  6. V. Karunakaran and N. V. Kalpakam, Boehmians representing measures, Houston J. Math. 26 (2000), no. 2, 377-386
  7. J. Mikusinski and T. K. Boehme, Operational calculus. Vol. II, International Series of Monographs in Pure and Applied Mathematics, 110. Pergamon Press, Oxford; PWN| Polish Scientific Publishers, Warsaw, 1987
  8. J. Mikusinski and P. Mikusinski, Quotients de suites et leurs applications dans l'analyse fonctionnelle, C. R. Acad. Sci. Paris Ser. I Math. 293 (1981), no. 9, 463-464
  9. P. Mikusinski, Convergence of Boehmians, Japan. J. Math. (N.S.) 9 (1983), no. 1, 159- 179 https://doi.org/10.4099/math1924.9.159
  10. P. Mikusinski, On harmonic Boehmians, Proc. Amer. Math. Soc. 106 (1989), no. 2, 447-449
  11. P. Mikusinski, Tempered Boehmians and ultradistributions, Proc. Amer. Math. Soc. 123 (1995), no. 3, 813-817
  12. P. Mikusinski and A. Zayed, The Radon transform of Boehmians, Proc. Amer. Math. Soc. 118 (1993), no. 2, 561-570
  13. D. Nemzer, Periodic Boehmians. II, Bull. Austral. Math. Soc. 44 (1991), no. 2, 271-278 https://doi.org/10.1017/S0004972700029713
  14. D. Nemzer, The dual space of ${\beta}({\Gammer})$, Internat. J. Math. Math. Sci. 20 (1997), no. 1, 111-114 https://doi.org/10.1155/S0161171297000161
  15. D. Nemzer, Generalized functions and an extended gap theorem, Indian J. Pure Appl. Math. 35 (2004), no. 1, 43-49
  16. D. Nemzer, Lacunary Boehmians, Integral Transforms Spec. Funct. 16 (2005), no. 5-6, 451-459
  17. W. Rudin, Functional analysis, Second edition. International Series in Pure and Applied Mathematics. McGraw-Hill, Inc., New York, 1991
  18. L. Schwartz, Theorie des distributions, Publications de l'Institut de Mathematique de l'Universite de Strasbourg, No. IX-X. Nouvelle edition, entierement corrigee, refondue et augmentee. Hermann, Paris, 1966

Cited by

  1. On the Generalized Krätzel Transform and Its Extension to Bohemian Spaces vol.2013, 2013, https://doi.org/10.1155/2013/841585
  2. On a Widder potential transform and its extension to a space of locally integrable Boehmians vol.18, 2015, https://doi.org/10.1016/j.jaubas.2014.03.004
  3. REAL COVERING OF THE GENERALIZED HANKEL-CLIFFORD TRANSFORM OF FOX KERNEL TYPE OF A CLASS OF BOEHMIANS vol.52, pp.5, 2015, https://doi.org/10.4134/BKMS.2015.52.5.1607
  4. An extension of certain integral transform to a space of Boehmians vol.17, 2015, https://doi.org/10.1016/j.jaubas.2014.02.003
  5. A class of Boehmians for a recent generalization of Hankel–Clifford transformation of arbitrary order vol.27, pp.5-6, 2016, https://doi.org/10.1007/s13370-015-0382-z