Consider the classical Schwartz space $\mathcal{S}(\mathbb{R})$ together with the Fourier transform $\mathcal{F} : \mathcal{S}(\mathbb{R}) \rightarrow \mathcal{S}( \mathbb{R})$.

Consider the subspace $V$ of the even, smooth functions on the interval $[-1,1]$.

Can you construct a (bounded) operator $D:\mathcal{S}(\mathbb{R}) \rightarrow \mathcal{S}(\mathbb{R}) $ such that $$ D \mathcal{F} v = 0, \quad Dv=v \qquad\forall v \in V ?$$ Observe that by Paley-Wiener, the intersection $\mathcal{F}V \cap V =0$ is trivial. What is the associated Schwartz kernel?