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The Fourier projection mappings $\{ P_N \}$ form an equicontinuous family of linear maps on $E'(S^1)$ as well?

Let $S^1=\mathbb{R}/\mathbb{Z}$ and define the Fourier projection operator $P_N$ for each $N \in \mathbb{N}$ as \begin{equation} P_N(f)=\sum_{n=-N}^N \langle f, e_n \rangle_{L^2} e_n \end{equation} ...