Frames in Hilbert Spaces

Show that $\left\| x_{n}\right\|^{2} \leq A$ for every $n \in \mathbb {N}$.

Show that if $\left\| x_{m}\right\|^{2} < A$ for some $m$, then $\left\{ x_{n}\right\}_{n \neq m}$ is a frame for $H$, and the optimal lower frame bound for $\left\{ x_{n}\right\}_{n \neq m}$ is $A-\left\| x_{m}\right\|^{2}$ (note that this new frame might not be tight).

Show that if $\left\| x_{m}\right\|^{2} = A$ for some $m$, then $x_{m} \perp x_{n}$ for all $n \neq m$.

$\left\| x_{n}\right\|^{2} = A$ for every $n$.

$\left\{ x_{n}\right\}$ is an orthogonal (but not necessarily orthonormal) sequence with no zero elements.

$\left\{ x_{n}\right\}$ is a basis for $H$.

$\left\{ x_{n}\right\}$ is $\omega$-independent.

$\left\{ x_{n}\right\}$ is an exact frame.

Use these equivalences to show that $\left\{ x_{n}\right\}$ is an exact Parseval frame for $H$ if and only if it is an orthonormal basis for $H$.

Show that if $b > 1$, then $\overline{\operatorname {span}}\left\{ e_{b n}\right\}_{n \in \mathbb {Z}}$ is a proper subspace of $L^{2}(\mathbf{T})$. In particular, find a nonzero function in $L^{2}(\mathbf{T})$ that is orthogonal to $e_{b n}$ for every $n \in \mathbb {Z}$.

Show that if $0 < b < 1$ then $\left\{ e_{b n}\right\}_{n \in \mathbb {Z}}$ is a tight frame for $L^{2}(\mathbf{T})$, and find the frame bound. Show directly that this sequence is not orthogonal, and demonstrate that it is not a basis by finding two distinct ways to write the constant function as $1 = \sum c_{n} e_{b n}$, where the series converges in $L^{2}$-norm. Show that $\left\{ e_{b n}\right\}_{n \neq 0}$ is a frame for $L^{2}(\mathbf{T})$, and therefore $\left\{ e_{b n}\right\}_{n \in \mathbb {Z}}$ is inexact.

Prove the following perturbation result for frames. Suppose that $\left\{ x_{n}\right\}$ is a frame for a Hilbert space $H$ with frame bounds $A, B$, and $\left\{ y_{n}\right\} \subseteq H$ is such that $\left\{ f_{n}-g_{n}\right\}$ is a Bessel sequence with Bessel bound $K$. Show that $\left\{ g_{n}\right\}$ is a frame if $K < A$.

Show that if $\left\{ h_{n}\right\}$ is a sequence in $H$ that satisfies $K = \sum \left\| h_{n}\right\|^{2} < \infty$, then $\left\{ h_{n}\right\}$ is a Bessel sequence with Bessel bound $K$.

Exercise 8.9 showed that $\left\{ e^{2 \pi i b n t}\right\}_{n \in \mathbb {Z}}$ is a frame for $L^{2}(\mathbf{T})$ when $0 < b \leq 1$. Combine this with parts (a) and (b) of this exercise to formulate and prove a theorem establishing a sufficient condition on numbers $\lambda_{n} \in \mathbb {R}$ so that $\left\{ e^{2 \pi i \lambda_{n} t}\right\}_{n \in \mathbb {Z}}$ is a frame for $L^{2}(\mathbf{T})$. Do you think your result is optimal?

$\left\{ v_{1}, \ldots , v_{n}\right\}$ is a Bessel sequence in $\mathbf{F}^{d}$, its synthesis operator corresponds to multiplication by the matrix $R$ that has $v_{1}, \ldots , v_{n}$ as columns, and its analysis operator is $C = R^{*}$, the Hermitian of the matrix $R$.

The following are equivalent: (i) $\left\{ v_{1}, \ldots , v_{n}\right\}$ spans $\mathbf{F}^{d}$, (ii) $S = C^{*} C$ is positive definite, (iii) $\left\{ v_{1}, \ldots , v_{n}\right\}$ is a frame for $\mathbf{F}^{d}$. Further, in case these hold, the optimal frame bounds for $\left\{ v_{1}, \ldots , v_{n}\right\}$ are $\lambda_{1}, \lambda_{d}$, where $\lambda_{1}$ is the smallest eigenvalue of $S$ and $\lambda_{d}$ is the largest eigenvalue.

The following are equivalent: (i) $\left\{ v_{1}, \ldots , v_{n}\right\}$ is linearly independent, (ii) $G = C C^{*}$ is positive definite, (iii) $\left\{ v_{1}, \ldots , v_{n}\right\}$ is a Riesz sequence in $\mathbf{F}^{d}$.

Show that $\left\{ v_{1}, \ldots , v_{N}\right\}$ is a frame sequence, i.e., it is a frame for $\operatorname {span}\left\{ v_{1}, \ldots , v_{N}\right\}$.

Assume $\left\{ v_{1}, \ldots , v_{N}\right\}$ is linearly independent, and let $A, B$ be frame bounds for $\left\{ v_{1}, \ldots , v_{N}\right\}$ as a frame for its span. Show that

Show that if $U, V \in \mathcal{B}(H)$ are positive and $U \leq V$ then $\left\| U\right\| \leq \left\| V\right\|$.

Let $S$ be the frame operator for a frame $\left\{ x_{n}\right\}$ that has frame bounds $A, B$. Prove the operator inequalities

and

and use either one of these to show that $S$ is a topological isomorphism.

If $\left\{ y_{n}\right\}$ is a Bessel sequence in $H$ that is an alternative dual for $\left\{ x_{n}\right\}$, then $\left\{ y_{n}\right\}$ is a frame and $\left\{ x_{n}\right\}$ is an alternative dual frame for $\left\{ y_{n}\right\}$.

A Bessel sequence $\left\{ y_{n}\right\}$ is the canonical dual frame for $\left\{ x_{n}\right\}$ if and only if it is an alternative dual and $\operatorname {range}(C) = \operatorname {range}(\widetilde{C})$, where $\widetilde{C}$ is the analysis operator of $\left\{ y_{n}\right\}$.

$\left\{ P x_{n}\right\}$ is a frame for $M$ with the same frame bounds as $\left\{ x_{n}\right\}$, and $\left\{ P \widetilde{x}_{n}\right\}$ is an alternative dual frame for $\left\{ P x_{n}\right\}$.

$\left\{ P \widetilde{x}_{n}\right\}$ is the canonical dual frame for $\left\{ P x_{n}\right\}$ if and only if $PS = SP$, where $S$ is the frame operator for $\left\{ x_{n}\right\}$.

Show that $\widetilde{C} = C S^{-1}$ and $\widetilde{R} = S^{-1} R$.

Show that the orthogonal projection $P$ of $\ell^{2}$ onto range $(C)$ is $P c = C \widetilde{R} c = C S^{-1} R c = \left\{ \left\langle \sum_{n} c_{n} \widetilde{x}_{n}, x_{k} \right\rangle \right\}_{k \in \mathbb {N}}, \quad c = \left(c_{n}\right) \in \ell^{2}$.

$\left\{ y_{n}\right\}$ is an alternative dual frame for $\left\{ x_{n}\right\}$.

There is a bounded left-inverse $V$ of $C$ such that $y_{n} = V \delta_{n}$ for all $n \in \mathbb {N}$ (that is, $V: \ell^{2} \rightarrow H$ is bounded, $V C = I$, and $y_{n} = V \delta_{n}$).

$\left\{ x_{n}\right\}$ is norm-bounded above, with $\left\| x_{n}\right\|^{2} \leq B$ for every $n$.

If $\left\{ x_{n}\right\}$ is exact then it is norm-bounded below, with $A \leq \left\| x_{n}\right\|^{2}$ for every $n$.

Show that $\operatorname {range}(C) = \operatorname {range}(\widetilde{C})$.

Show that the pseudoinverse $R^{\dagger }$ of the synthesis operator $R$ is $R^{\dagger } = \widetilde{C}$, so

Show that the optimal frame bounds for $\left\{ x_{n}\right\}$ are

Show that $\left\{ x_{n}\right\}$ is complete if and only if $R$ has dense range.

Show that $\left\{ x_{n}\right\}$ is $\ell^{2}$-independent if and only if $R$ is injective.

Now assume that $\left\{ x_{n}\right\}$ is a Bessel sequence that is a conditional Schauder basis for $H$ (see Exercise 7.1 for an example). Show that there exists some $x \in H$ such that $\left( \left\langle x, \widetilde{x}_{n} \right\rangle \right) \notin \ell^{2}$, where $\left(\widetilde{x}_{n}\right)$ is the dual basis to $\left\{ x_{n}\right\}$. Show that the sequence $\left\{ x_{n}\right\} \cup \left\{ x\right\}$ is $\ell^{2}$-independent but not $\omega$-independent.

$\left\{ x_{n}\right\}$ is a frame for $H$ if and only if there exists a closed subspace $M \subseteq \ell^{2}$ and a topological isomorphism $T: M \rightarrow H$ such that $x_{n} = T P_{M} \delta_{n}$ for every $n$.

$\left\{ x_{n}\right\}$ is a Parseval frame for $H$ if and only if the operator $T$ in part (a) can be taken to be an isometric isomorphism.

$\left\{ x_{n}\right\}$ is a frame for $H$ if and only if there exists a Hilbert space $K \supseteq H$ and a Riesz basis $\left\{ e_{n}\right\}$ for $K$ such that $P_{H} e_{n} = x_{n}$ for each $n$.

$\left\{ x_{n}\right\}$ is a Parseval frame for $H$ if and only if there exists a Hilbert space $K \supseteq H$ and an orthonormal basis $\left\{ e_{n}\right\}$ for $K$ such that $P_{H} e_{n} = x_{n}$ for each $n$.

Exhibit a frame for $\mathbf{F}^{d}$ that contains infinitely many nonzero vectors.

Show that if $\left\{ x_{n}\right\}_{n \in I}$ is a frame for $\mathbf{F}^{d}$ and $\inf_{n \in I}\left\| x_{n}\right\| > 0$, then $I$ is finite.

Show that $\left\{ x_{n}\right\}$ and $\left\{ y_{n}\right\}$ are equivalent if and only if range $\left(C_{X}\right) =$ range $\left(C_{Y}\right)$, where $C_{X}, C_{Y}$ are the analysis operators for $\left\{ x_{n}\right\}$ and $\left\{ y_{n}\right\}$.

Show that part (a) is equivalent to:

Exhibit a frame $\left\{ x_{n}\right\}$ and a permutation $\sigma$ of $\mathbb {N}$ that fixes all but finitely many elements of $\mathbb {N}$ such that $\left\{ x_{n}\right\}$ is not equivalent to $\left\{ x_{\sigma (n)}\right\}$. Show that this cannot happen if $\left\{ x_{n}\right\}$ is an exact frame.

Exhibit a frame $\left\{ x_{n}\right\}$ and a sequence of signs $\varepsilon_{n} = \pm 1$ such that $\left\{ x_{n}\right\}$ is not equivalent to $\left\{ \varepsilon_{n} x_{n}\right\}$. Show that this cannot happen if $\left\{ x_{n}\right\}$ is an exact frame.

Show that if $\operatorname {range}\left(C_{X}\right) \perp \operatorname {range}\left(C_{Y}\right)$ then $\left\{ \left(x_{n}, y_{n}\right)\right\}$ is a frame for $H \times K$, and it is a Parseval frame if and only if both $\left\{ x_{n}\right\}$ and $\left\{ y_{n}\right\}$ are Parseval. (When $\operatorname {range}\left(C_{X}\right) \perp \operatorname {range}\left(C_{Y}\right)$, we say that $\left\{ x_{n}\right\}$ and $\left\{ y_{n}\right\}$ are orthogonal frames.)

Suppose that, in addition, we have $\operatorname {range}\left(C_{X}\right) \oplus \operatorname {range}\left(C_{Y}\right) = \ell^{2}$. Show that $\left\{ \left(x_{n}, y_{n}\right)\right\}$ is a Riesz basis for $H \times K$, and it is an orthonormal basis if and only if $\left\{ x_{n}\right\}$ and $\left\{ y_{n}\right\}$ are each Parseval frames (in this case, we call $\left\{ \left(x_{n}, y_{n}\right)\right\}$ a superframe).

Give examples of frames $\left\{ x_{n}\right\}$ and $\left\{ y_{n}\right\}$ that are not Riesz bases but are such that $\left\{ \left(x_{n}, y_{n}\right)\right\}$ is a Riesz basis for $H \times H$.

Given a frame $\left\{ x_{n}\right\}$ for $H$, show there exists a frame $\left\{ y_{n}\right\}$ for $H$ such that $\left\{ \left(x_{n}, y_{n}\right)\right\}$ is a Riesz basis for $H \times H$.

Let $\left\{ x_{n}\right\} ,\left\{ y_{n}\right\}$ be frames for Hilbert spaces $H, K$, respectively. Show that the tensor product sequence $\left\{ x_{m} \otimes y_{n}\right\}_{m, n \in \mathbb {N}}$ is a frame for $H \otimes K = \mathcal{B}_{2}(H, K)$ (see Appendix B for definitions).

Let $E, F$ be measurable subsets of $\mathbb {R}$. Show that if $\left\{ f_{n}\right\}$ is a frame for $L^{2}(E)$ and $\left\{ g_{n}\right\}$ is a frame for $L^{2}(F)$, then $\left\{ \overline{f_{m}(x)} g_{n}(y)\right\}_{m, n \in \mathbb {N}}$ is a frame for $L^{2}(E \times F)$.

$\left\{ x_{n}\right\}_{n \neq n_{k}}$ is complete (and hence a frame) for each $k \in \mathbb {N}$, and there exists a single constant $L > 0$ that is a lower frame bound for each frame $\left\{ x_{n}\right\}_{n \neq n_{k}}$.

$\sup_{k} \left\langle x_{n_{k}}, \widetilde{x}_{n_{k}} \right\rangle < 1$.

Show further that if these statements hold, then there exists an infinite subsequence $\mathcal{G}_{\varepsilon }$ of $\mathcal{G}$ such that $\mathcal{F} \backslash \mathcal{G}_{\varepsilon }$ is a frame for $H$ with frame bounds $L(A / B)-\varepsilon , 1$.

$U\left(\operatorname {ker}(C)\right) \subseteq \operatorname {ker}(C)$.

$V^{*}\left(\operatorname {ker}(R)\right) \subseteq \operatorname {ker}(R)$.

$V\left(\operatorname {range}(C)\right) \subseteq \operatorname {range}(C)$.

$U^{*}\left(\operatorname {range}(R)\right) \subseteq \operatorname {range}(R)$.

Prove that the operators $U, V, U^{*}$, and $V^{*}$ appearing in Example 8.49 have no eigenvalues.

Show that if $0 < b < 1$, then there is an infinite set $J \subseteq \mathbb {Z}$ such that $\left\{ e^{2 \pi i b n t}\right\}_{n \notin J}$ is a frame for $L^{2}(\mathbf{T})$.