Let $\mathcal{S}\subset \mathcal{H}=L^2(\mathbb{R}^n)\subset \mathcal{S}^*$ be the Schwartz space, the Hilbert space and the space of tempered distributions respectively. Consider an algebra $\mathcal{A}$ of operators with continuous spectrum. We see delta function as the element of $\mathcal{S}^*$, i.e. continuous (in kernal topology with seminorms $f \mapsto \|Af\|, A\in \mathcal{A}, f\in \mathcal{H}$) linear functional on $\mathcal{S}$.
Question: how can we correctly define an action of $\delta(A-\vec{x}), A\in \mathcal{A}, \vec{x}\in \mathbb{R}^n$ on $f\in \mathcal{H}\,?$
For example, here we have the algebra $\mathcal{A}$ generated by operators $X \colon f(\vec{x})\mapsto \vec{x} f(\vec{x})$ and $P \colon f(\vec{x})\mapsto \partial_\vec{x} f(\vec{x})$ with $[X,P]=iI$. Emilio Pisanty tell that functions of the position operator should work $⟨x|F(X)=F(\vec{x})⟨x|$ but I don't understand how it work with delta function.
Also we can find (here on p.13 formula (74)) the Fourier transform of operator $\rho(\vec{x})=\delta(X-\vec{x})$: $$\rho(\vec{k})=\int\limits_{\mathbb{R^n}} e^{-i\vec{k}\cdot \vec{x}}\rho(\vec{x})\,d\vec{x}=e^{-i\vec{k}X}.$$
I will be very grateful for any remarks and comments.