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Abstract

We prove precise decomposition results and logarithmically convex estimates in certain weighted spaces of holomorphic germs near $\mathbb{R}$. These imply that the spaces have a basis and are tamely isomorphic to the dual of a power series space of finite type which can be calculated in many situations. Our results apply to the Gelfand–Shilov spaces $S^1_{\alpha}$ and $S_1^{\alpha}$ for $\alpha>0$ and to the spaces of Fourier hyperfunctions and of modified Fourier hyperfunctions.