Suppose $f_\lambda(x),\lambda \in \mathbb{R}$ is a continuous/smooth family of bounded function on $\mathbb{R}$. Given a measurable set $\mathfrak{m}$ on $\mathbb{R}$, define $$E_\mathfrak{m}:L^2\longrightarrow C(\mathbb{R})$$ by $$E_\mathfrak{m}(\varphi):=\int_\mathfrak{m}f_\lambda\langle f_\lambda,\varphi\rangle d\;\lambda.$$
Is there any condition we can put on $f_\lambda$ so that $E$ become a resolution of the identity?
Is it the case that so long as $f_\lambda(x)=\exp(i \lambda x)$ when $|x|>>0$, and $f_\lambda$ is separating family, then we always get a resolution of the identity?
