2025/09/14 by Philipp J. di Dio, di Dio, Philipp J.
Mathematics · Decision Sciences · #Fuzzy Systems and Optimization #Fuzzy and Soft Set Theory #Advanced Banach Space Theory
paper · pdf · doi:10.48550/arxiv.2509.11339
In 1989, A. J. Duran [Proc. Amer. Math. Soc. 107 (1989), 731-741] showed, that for every complex sequence (sα)α∈ℕ0n there exists a Schwartz function f\inS(ℝn,ℂ) with supp f⊆ [0,∞)n such that sα= ∫ xα⋅ f(x)~dx for all α∈ℕ0n. 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,∞)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.