Recall that in many Fourier Analysis texts, given a function $\Psi$ such that $\hat{\Psi}\in\mathcal{D}(\mathbb R^d)$, $\hat\Psi\ge0$ is supported in an annulus, and $\sum_{j\in\mathbb Z}\hat\Psi(2^j\xi)=1$ for all $\xi\ne0$, we can define the Littlewood-Paley operator $$\Delta_j(f) := \left(\hat\Psi(2^{-j}\xi)\hat{f}(\xi)\right)^\vee$$ for $f\in\mathscr{S}'(\mathbb R^d)$. Furthermore, if we set $$\hat\Phi(\xi)=1-\sum_{j\ge0}\Psi(2^{-j}\xi)$$ then we can define $$\dot S_j(f):=\left(\hat\Phi(2^{-j}\xi)\hat f(\xi)\right)^\vee$$ and using this, we define $\mathscr{S}'_h(\mathbb R^d)$ to be the space of tempered distributions $u$ such that $\lim\limits_{j\to-\infty}\dot S_j(u)=0$ in $\mathscr{S}'(\mathbb R^d)$.

Now, clearly the space $\mathscr{P}$ of polynomials in $\mathscr{S}'$ intersects trivially with $\mathscr{S}_h'$, but is $\mathscr{S}_h'$ in fact a complementary subspace for $\mathscr P$? It would suffice to show that the limit $T(u):=\lim\limits_{j\to-\infty}\dot S_j(u)$ exists for any $u\in\mathscr{S}'$, since $T(u)\in\mathscr P$ and $T^2=T$ wherever $T$ is defined, which would make $T$ into a projection. However, I have only managed to show that $T(u)$ is defined when $\operatorname{ord}u=0$, i.e. when $u$ is locally a measure. But is $\mathscr{S}'_h$ even a complementary subspace, and if so, how do we show that?


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