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

Question

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} where $e_n(x)=e^{inx}$, $f : S^1 \to \mathbb{C}$ is a smooth function and $\langle , \rangle_{L^2}$ is the $L_2$ inner product.

Let us denote the Frechet space of smooth functions on $S^1$ as $\mathcal{E}(S^1)$. Then, I know that $\{P_N\}_{N=1}^\infty$ form a equicontinuous family of linear maps on $\mathcal{E}(S^1)$.

Next, consider the space of periodic distiributions $\mathcal{E}'(S^1)$ equipped with strong dual topology. Then, the domain $P_N$ may be extended to $\mathcal{E}'(S^1)$ via dual pairing.

My question is whether or not $\{P_N \}_{N=1}^\infty$ form a equicontinuious family of linear maps on $\mathcal{E}'(S^1)$ as well. The generalized definition of equicontinuity is as follows:

https://en.wikipedia.org/wiki/Equicontinuity#Equicontinuous_linear_maps

I strongly believe that the projection still remain as equicontinuous on $\mathcal{E}'(S^1)$, but cannot justify this myself. Could anyone please help me or recommend any references?

Answer 1

By "extended to $\mathcal E'(S^1)$ via dual pairing", I assume you mean that the extension of $P_N:\mathcal E(S^1)\to\mathcal E(S^1)$ to $\mathcal E'(S^1)$ is given by its transpose ${}^t\mkern-2mu P_N:\mathcal E'(S^1)\to\mathcal E'(S^1)$.

There are several ways of proving the equicontinuity of $\lbrace\mkern1mu{}^t\mkern-2mu P_N\rbrace_{N=1}^\infty$, some more high-tech, some more elementary. The way shown below is rather elementary.

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There are mainly two ingredients:

• from the equicontinuity of $\lbrace P_N\rbrace_{N=1}^{\infty}$ on $\mathcal E(S^1)$, it is easy to check that if $B$ is a bounded subset of $\mathcal E(S^1)$, then $\bigcup_{N=1}^\infty P_N(B)$ is also a bounded subset of $\mathcal E(S^1)$,
• from the definition of the strong dual topology, the polars in $\mathcal E'(S^1)$ of bounded subsets of $\mathcal E(S^1)$ form a basis of the filter of neighbourhoods of the origin in $\mathcal E'(S^1)$.

Then, it is easy to check that if $V$ is the polar in $\mathcal E'(S^1)$ of a bounded subset $B$ of $\mathcal E(S^1)$ and $U$ is the polar in $\mathcal E'(S^1)$ of $\bigcup_{N=1}^\infty P_N(B)$, then $(\mkern2mu{}^t\mkern-2mu P_N)(U)\subseteq V$ for every $N\geq 1$: indeed, $$|\langle\mkern2mu{}^t\mkern-2mu P_N(T),f\rangle|=|\langle T,P_N(f)\rangle|\leq 1$$ for every $N\geq 1$, $T\in U$ and $f\in B$.