Absolutely continuous representing measures of complex sequences

Abstract

In 1989, A. J. Duran [Proc. Amer. Math. Soc. 107 (1989), 731-741] showed, that for every complex sequence $(s_{\hat{I}}\pm {)}_{\hat{I}\pm \in {\mathbb{N}}_0^n}$ there exists a Schwartz function $f\in \mathcal{S}({\mathbb{R}}^n,\mathbb{C})$ with $\mathrm{supp}f\subseteq [0,\infty {)}^n$ such that $s_{\hat{I}}\pm =\int x^{\hat{I}}\pm \cdot f(x)\mathrm{d}x$ for all $\hat{I}\pm \in {\mathbb{N}}_0^n$. It has been claimed to be a generalization of the result by T. Sherman [Rend. Circ. Mat. Palermo 13 (1964), 273-278], that every complex sequences is represented by a complex measure on $[0,\infty {)}^n$. In the present work we use the convolution of sequences and measures to show, that Duran's result is a trivial consequence of Sherman's result. We use our easy proof to extend the Schwartz function result and to show the flexibility in choosing very specific functions $f$.