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I am interested in a generalization of the following finite-dimensional results in infinite dimensional vector-space with nuclear structure, especially for the cases of the spaces of distributions $\...

For a given function $f\in L^1(\mathbb{R})$, suppose that the $$\check{f}(x)=\int_\mathbb{R} \hat{f}(\zeta)e^{2\pi i\zeta x}d\zeta$$ almost every where converges in $\mathbb{R}$. Then, can we say that ...