THE PRODUCT OF ANALYTIC FUNCTIONALS IN Z'

Title & Authors
THE PRODUCT OF ANALYTIC FUNCTIONALS IN Z'
Li, Chenkuan; Zhang, Yang; Aguirre, Manuel; Tang, Ricky;

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 $\small{F(N_m)}$ for $\small{\delta^{(m)}(s)}$ 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 $\small{\delta^{(\alpha)}(s)}$ for $\small{\alpha}$>$\small{0}$.
Keywords
Paley-Wiener-Schwartz theorem$\small{\delta}$-function;product;fractional derivative and multiplier space;
Language
English
Cited by
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

3.
M. A. Aguirre T, The series expansion of $\delta^{(k)}$(r - c), Math. Notae 35 (1991), 53-61

4.
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

5.
M. A. Aguirre T, The expansion of $\delta^{(k-1)}(m^2+P)$, Integral Transform. Spec. Funct. 8 (1999), no. 1-2, 139-148

6.
P. Antosik, J. Mikusinski, and R. Sikorski, Theory of Distributions. The Sequential Approach, Elsevier Scientific Publishing Co., Amsterdam; PWN-Polish Scientific Publishers, Warsaw, 1973

7.
H. J. Bremermann, Distributions, Complex Variables, and Fourier Transforms, Addison-Wesley Publishing Co., Inc., Reading, Mass.-London 1965

8.
L. Z. Cheng and C. K. Li, A commutative neutrix product of distributions on $R^m$, Math. Nachr. 151 (1991), 345-355

9.
B. Fisher, The product of distributions, Quart. J. Math. Oxford Ser. (2) 22 (1971), 291-298

10.
B. Fisher, A noncommutative neutrix product of distributions, Math. Nachr. 108 (1982), 117-127

11.
B. Fisher and K. Tas, The convolution of functions and distributions, J. Math. Anal. Appl. 306 (2005), no. 1, 364-374

12.
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

13.
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

14.
B. Fisher and K. Tas, On the commutative product of distributions, J. Korean Math. Soc. 43 (2006), no. 2, 271-281

15.
S. Gasiorowicz, Elementary Particle Physics, J. Wiley and Sons, Inc., N. Y. 1966

16.
I. M. Gel'fand and G. E. Shilov, Generalized functions, Vol. I, Academic Press, 1964

17.
A. Kilicman and B. Fisher, Further results on the noncommutative neutrix product of distributions, Serdica 19 (1993), no. 2-3, 145-152

18.
B. H. Li, Non-standard analysis and multiplication of distributions, Sci. Sinica 21 (1978), no. 5, 561-585

19.
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

20.
C. K. Li, The products on the unit sphere and even-dimension spaces, J. Math. Anal. Appl. 305 (2005), no. 1, 97-106

21.
C. K. Li, An approach for distributional products on $R^m$, Integral Transforms Spec. Funct. 16 (2005), no. 2, 139-151

22.
C. K. Li and B. Fisher, Example of the neutrix product of distributions on $R^m$, Rad. Mat. 6 (1990), no. 1, 129-137

23.
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

24.
L. Schwartz, Theorie des distributions a valeurs vectorielles, I, II, Ann. Inst. Fourier, Grenoble 7,8 (1957/58), 1-141. (1-209)

25.
F. Treves, Topological vector spaces, distributions and kernels, Academic Press, 1970