Uncertainty principle in harmonic analysis

Let $P, M$ be the position and momentum operators introduced in equation (11.26). These operators are not defined on all of $L^{2}(\mathbb {R})$. Instead, define domains

which are dense subspaces of $L^{2}(\mathbb {R})$. Restricted to these domains, $P$ maps $D_{P}$ into $L^{2}(\mathbb {R})$ and $M$ maps $D_{M}$ into $L^{2}(\mathbb {R})$. Show that $P$ and $M$ are unbounded even when restricted to these domains, i.e.,

The commutator of position and momentum is the operator $[P, M] = P M-M P$. Show that $[P, M] = -\frac{1}{2 \pi i} I$ in the sense that $[P, M] f = -\frac{1}{2 \pi i} f$ for all differentiable functions $f$. How does this relate to equation (11.32)?

This exercise will give an abstract operator-theoretic version of the Uncertainty Principle.

Let $S$ be a subspace of a Hilbert space $H$, and let $A, B: S \rightarrow H$ be linear but possibly unbounded operators. By replacing $S$ with the smaller space domain $(A B) \cap$ domain $(B A)$ if necessary, we may assume that $A, B, A B$, and $B A$ are all defined on $S$.

Apply Exercises $11.29-11.31$ to the position and momentum operators $P$ and $M$ to derive the Classical Uncertainty Principle,

for $g \in \mathcal{S}(\mathbb {R})$.

Remark An extension by density argument can be used to prove that equation (11.33) extends to all $g \in L^{2}(\mathbb {R})$, or integration by parts for absolutely continuous functions can be used to prove directly that equation (11.33) holds whenever $\left\| x g(x)\right\|_{L^{2}}\left\| \xi \widehat{g}(\xi )\right\|_{L^{2}}$ is finite, see [Heil].

Modify Battle's argument to prove the Weak $B L T$ : If $\mathcal{G}(g, 1,1)$ is a Riesz basis for $L^{2}(\mathbb {R})$ then

where $\widetilde{g}$ is the dual system generator from Theorem 11.31(b).

Remark It requires some work, but it can be shown that the Weak BLT implies Theorem 11.35; see [DJ93] or the survey paper [BHW95].

Suppose $f$ is a real, continuously differentiable function on [a, b], $f(a) = f(b) = 0$, and

Prove that

and that

Assume $f$ is a measurable function on $(-\infty , \infty )$ such that $\left|f(x)\right| < e^{-\left|x\right|}$ for all $x$. Prove that its Fourier transform $\hat{f}$ cannot have compact support, unless $f(x) = 0$ a.e.

In the section we claimed that

The point of this exercise is to verify the middle equality, and hence all three equalities. Note that

Use integration by parts to show that this equals $i w \hat{f}(w)$, and hence that the equation holds.

Show that (6.8) (in the text of the section) is true whenever $\alpha^{2}+\beta^{2} \leq 1$.