Weighted function spaces: Convolutors, multipliers, and mollifiers

We study smooth function spaces of Gelfand–Shilov type, with global behavior governed through a translation‐invariant Banach function space (TIBF) and localized via a weight function system. We clarify the roles of the TIBF, convolution, and pointwise multiplication in connection with the weight function system. Our primary goal is to characterize these function spaces—as well as the corresponding convolutor and multiplier spaces—through mollification. For this purpose, we introduce the moment‐wise decomposition factorization property for pairs of compactly supported smooth functions, and establish complete characterizations in terms of mollifications with these windows.