DOI QR코드

DOI QR Code

THE PRODUCT OF ANALYTIC FUNCTIONALS IN Z'

  • Li, Chenkuan (DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE BRANDON UNIVERSITY) ;
  • Zhang, Yang (DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE BRANDON UNIVERSITY) ;
  • Aguirre, Manuel (NUCLEO CONSOLIDADO DE MATEMATICA PURA Y APLICADA FACULTAD DE CIENCIAS EXACTAS) ;
  • Tang, Ricky (DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE BRANDON UNIVERSITY)
  • Published : 2008.03.31

Abstract

Current studies on products of analytic functionals have been based on applying convolution products in D' and the Fourier exchange formula. There are very few results directly computed from the ultradistribution space Z'. The goal of this paper is to introduce a definition for the product of analytic functionals and construct a new multiplier space $F(N_m)$ for $\delta^{(m)}(s)$ in a one or multiple dimension space, where Nm may contain functions without compact support. Several examples of the products are presented using the Cauchy integral formula and the multiplier space, including the fractional derivative of the delta function $\delta^{(\alpha)}(s)$ for $\alpha>0$.

References

  1. M. A. Aguirre T, A convolution product of (2j)th derivative of Dirac's delta in r and multiplicative distributional product between $r^{-k}$ and $\nabla(\Delta^j\delta)$, Int. J. Math. Math. Sci. (2003), no. 13, 789-799
  2. M. A. Aguirre T, The expansion in series (of Taylor types) of (k - 1) derivative of Dirac's delta in $m^2$ + P, Integral Transforms Spec. Funct. 14 (2003), no. 2, 117-127 https://doi.org/10.1080/10652460290029653
  3. M. A. Aguirre T, The series expansion of $\delta^{(k)}$(r - c), Math. Notae 35 (1991), 53-61
  4. M. A. Aguirre T, The expansion of $\delta^{(k-1)}(m^2+P)$, Integral Transform. Spec. Funct. 8 (1999), no. 1-2, 139-148 https://doi.org/10.1080/10652469908819222
  5. P. Antosik, J. Mikusinski, and R. Sikorski, Theory of Distributions. The Sequential Approach, Elsevier Scientific Publishing Co., Amsterdam; PWN-Polish Scientific Publishers, Warsaw, 1973
  6. H. J. Bremermann, Distributions, Complex Variables, and Fourier Transforms, Addison-Wesley Publishing Co., Inc., Reading, Mass.-London 1965
  7. L. Z. Cheng and C. K. Li, A commutative neutrix product of distributions on $R^m$, Math. Nachr. 151 (1991), 345-355 https://doi.org/10.1002/mana.19911510124
  8. B. Fisher, The product of distributions, Quart. J. Math. Oxford Ser. (2) 22 (1971), 291-298 https://doi.org/10.1093/qmath/22.2.291
  9. B. Fisher, A noncommutative neutrix product of distributions, Math. Nachr. 108 (1982), 117-127 https://doi.org/10.1002/mana.19821080110
  10. B. Fisher and K. Tas, The convolution of functions and distributions, J. Math. Anal. Appl. 306 (2005), no. 1, 364-374 https://doi.org/10.1016/j.jmaa.2005.01.004
  11. B. Fisher and K. Tas, On the composition of the distributions $x^\lambda_+$ and $x^\mu_+$, J. Math. Anal. Appl. 318 (2006), no. 1, 102-111 https://doi.org/10.1016/j.jmaa.2005.05.022
  12. B. Fisher and K. Tas, On the non-commutative neutrix product of the distributions $x^{r}ln^{p}$|x| and $x^{-s}$, Integral Transforms Spec. Funct. 16 (2005), no. 2, 131-138 https://doi.org/10.1080/1065246042000272018
  13. B. Fisher and K. Tas, On the commutative product of distributions, J. Korean Math. Soc. 43 (2006), no. 2, 271-281 https://doi.org/10.4134/JKMS.2006.43.2.271
  14. S. Gasiorowicz, Elementary Particle Physics, J. Wiley and Sons, Inc., N. Y. 1966
  15. I. M. Gel'fand and G. E. Shilov, Generalized functions, Vol. I, Academic Press, 1964
  16. A. Kilicman and B. Fisher, Further results on the noncommutative neutrix product of distributions, Serdica 19 (1993), no. 2-3, 145-152
  17. C. K. Li, The product of $r^{-k}$ and $\nabla\delta$ on $R^m$, Int. J. Math. Math. Sci. 24 (2000), no. 6, 361-369 https://doi.org/10.1155/S0161171200004233
  18. C. K. Li, The products on the unit sphere and even-dimension spaces, J. Math. Anal. Appl. 305 (2005), no. 1, 97-106 https://doi.org/10.1016/j.jmaa.2004.10.031
  19. C. K. Li, An approach for distributional products on $R^m$, Integral Transforms Spec. Funct. 16 (2005), no. 2, 139-151 https://doi.org/10.1080/1065246042000272117
  20. C. K. Li and B. Fisher, Example of the neutrix product of distributions on $R^m$, Rad. Mat. 6 (1990), no. 1, 129-137
  21. C. K. Li and E. L. Koh, The neutrix convolution product in Z'(m) and the exchange formula, Internat. J. Math. Math. Sci. 21 (1998), no. 4, 695-700 https://doi.org/10.1155/S0161171298000969
  22. L. Schwartz, Theorie des distributions a valeurs vectorielles, I, II, Ann. Inst. Fourier, Grenoble 7,8 (1957/58), 1-141. (1-209)
  23. F. Treves, Topological vector spaces, distributions and kernels, Academic Press, 1970
  24. M. A. Aguirre T, A generalization of convolution product of the distributional families related to the diamond operator, Thai J. Math. 2 (2004), 97-106
  25. B. H. Li, Non-standard analysis and multiplication of distributions, Sci. Sinica 21 (1978), no. 5, 561-585