Journal of the
Korean Mathematical Society JKMS

Current studies on products of analytic functionals have been based on applying convolution products in ${\mathcal D'}$ and the Fourier exchange formula. There are very few results directly computed from the ultradistribution space ${\mathcal Z'}$. The goal of this paper is to introduce a definition for the product of analytic functionals and construct a new multiplier space $F ({\mathcal N}_m)$ for $\delta^{(m)}(s)$ in a one or multiple dimension space, where ${\mathcal N}_m$ 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 $\alpha > 0$.

Keywords: Paley-Wiener-Schwartz theorem, $\delta$-function, product, fractional derivative and multiplier space

MSC numbers: Primary 46F10