Taylor expansion for generalized functions (Q1815465)

The author considers the following asymptotic Taylor expansion for a distribution \(f\in{\mathcal D}'\) on the straight line \(\{ hy;\;h\in{\mathbf R}\}, \;y\in{\mathbf R}^{n}\): \[ f(x+\varepsilon y)\sim\sum_{|k|=0}^{\infty} {{D^{k}f(x)}\over {k!}}(\varepsilon y)^{k},\;\text{as} \;\varepsilon\rightarrow 0\tag{1} \] introduced in [\textit{R. Estrada} and \textit{R. P. Kanwal}, Math. Methods Appl. Sci. 16, 297-304 (1993; Zbl 0796.46017)], which can be naturally extended to ultradistributions from \({\mathcal D}^{\prime\ast}\). The question of the author's interest is `what do one has to suppose on the distribution (or the ultradistribution) \(f\) that the asymptotic Taylor expansion (1) for \(f\) becomes its Taylor series?' The main result of the present paper gives the following answer to this question: The asymptotic Taylor expansion for \(f\in {\mathcal D} '\) (\(f\in {\mathcal D}^{\prime\ast}\)) on the straight line \(\{ hy;\;h\in {\mathbf R}\}\) is the Taylor series convergent in \({\mathcal D}'\) ( in \({\mathcal D}^{\prime\ast}\)) if and only if \(f\) is given by a real analytic function which can be extended as a holomorphic function on some region in \({\mathbf C}^{n}\). This paper is the extension of investigations in the cited above paper.