Let $F\in C(\mathbb{R}^d)$ with $F(x)\to 0$ as $|x|\to\infty$. Denote $P_jF$ the Littlewood-Paley projection to $|\xi|\sim 2^j$, i.e. $P_j F=\mathcal{F}^{-1}(\chi(2^{-j}\cdot)\mathcal{F}(F))$ for $\chi$ a smooth cut-off to an annulus such that $\sum_j \chi(2^{-j}\xi)=1$ for all $\xi\neq0$. Does it hold that $F=\sum_j P_j F$ in $\mathcal{S}'(\mathbb{R}^d)$?
I have tried 2 approaches but neither quite seems to work:
Plan of attack #1:
Since $F$ is decaying, there exist Schwarz functions $(F_n)_n$ with $F_n\to F$ in $L^\infty\cap \mathcal{S}'$. Let $\phi\in\mathcal{S}$. Then \begin{align*} \left|\left\langle\sum_j P_jF-\sum_jP_jF_n,\phi\right\rangle\right|=&\lim_{N\to\infty}\left|\left\langle \sum_{|j|\leq N}P_j(F-F_n),\phi\right\rangle\right|\\ \leq&\lim_{N\to\infty}\|F-F_n\|_\infty\left\|\sum_{|j|\leq N} P_j\phi\right\|_1 \end{align*} I then want to say that $\|\sum_{|j|\leq N} P_j\phi\|_1$ is uniformly bounded in $N$, to conclude that $\sum_j P_j F=\lim_n\sum_j P_jF_n$ in $\mathcal{S}'$. Then since $\sum_jP_j F_n=F_n\to F$ in $\mathcal{S}'$ we would be done. The problem is, in the case of $p=1$, I'm not sure a bound such as $\|\sum_{|j|\leq N} P_j\phi\|_1\lesssim\|\phi\|_1$ exists?
Plan of attack #2:
Since $\text{supp}\mathcal{F}(F-\sum_j P_j F)=\{0\}$ we know $F=p+\sum_j P_j F$ for some polynomial $p$. Since $F\to0$ as $\infty$, to show $p=0$ it suffices to show $\lim_{|x|\to\infty}(\sum_j P_jF(x))=0$. I can see that each $P_jF(x)\to_{|x|\to\infty}0$ using the expression of $P_jF$ as a convolution and the decay of $F$, however haven't been able to make much progress on the sum.
Disclaimer: this might not even be true, it is more that I think I have seen it used implicitly and it would be helpful if it was true...counterexamples also welcome!