Combining and decomposing frames

Abstract

We now give a list of ways in which two or more frames can be combined to obtain a new frame, for which the frame and its dual are related in a natural way to those of its constituent parts.

Appendices

Notes

The direct sum, and the associated notions of a lift and complement can be found throughout the frame literature. There can be some variation in terminology (when named), e.g., the term lift is used for the simple lift in [BF03].

Exercises

5.1.

Show that a frame $(f_j)$ is a disjoint union of tight frames if and only if each $f_j$ is an eigenvector of the frame operator S (such a frame is said to be semicritical ).

5.2.

Unions. Let $\varPhi =(\phi _j)$ and $\varPsi =(\psi _k)$ be frames for ${\mathscr {H}}_1$ and ${\mathscr {H}}_2$.

(a) Show that $\varPhi \cup \varPsi$, as a sequence in ${\mathscr {H}}_1\oplus {\mathscr {H}}_2$, has frame operator S, with

$$ S(f+g)=S_\varPhi (f)+S_\varPsi (g), \qquad \forall f\in {\mathscr {H}}_1,\ \forall g\in {\mathscr {H}}_2. $$

(b) Show that the dual frame is $\tilde{\varPhi }\cup \tilde{\varPsi }$, by using Exer. 5.4.

(c) Show the Gramian is block diagonal, i.e.,

$$ \mathrm{Gram}(\varPhi \cup \varPsi ) = \begin{pmatrix}\mathrm{Gram}(\varPhi )&{}0\\ 0&{}\mathrm{Gram}(\varPsi )\end{pmatrix}.$$

5.3.

Direct sums. Let $\varPhi =(\phi _j)_{j\in J}$ and $\varPsi =(\psi _j)_{j\in J}$ be finite frames for ${\mathscr {H}}_1$ and ${\mathscr {H}}_2$, with $V=[\phi _j]$, $W=[\psi _j]$. Show that $(\phi _j+\psi _j)$ is a frame for ${\mathscr {H}}_1\oplus {\mathscr {H}}_2$ if and only if

$$ \mathrm{ran}(V^*)\cap \mathrm{ran}(W^*)=0, $$

in which case

$$ \mathrm{ran}([\phi _j+\psi _j]^*)=\mathrm{ran}(V^*)+\mathrm{ran}(W^*) \qquad \hbox {(algebraic direct sum)}. $$

5.4.

Suppose $\varXi$ is a sequence in ${\mathscr {H}}_1\oplus {\mathscr {H}}_2$ for which $S=S_\varXi$ can be decomposed

$$\begin{aligned} S(f+g) = S_\varPhi (f) + S_\varPsi (g), \qquad \forall f\in {\mathscr {H}}_1, \ \forall g\in {\mathscr {H}}_2, \end{aligned}$$

(5.13)

where $\varPhi$ and $\varPsi$ are frames for ${\mathscr {H}}_1$ and ${\mathscr {H}}_2$, e.g., the sequences

$$\begin{aligned} \varPhi \cup \varPsi , \quad \varPhi \oplus \varPsi , \quad \varPhi \mathbin {\hat{+}}\varPsi . \end{aligned}$$

(5.14)

(a) Show that $\varXi$ is a frame for ${\mathscr {H}}_1\oplus {\mathscr {H}}_2$, with frame bounds

$$ A = \min \{A_\varPhi ,A_\varPsi \}, \qquad B = \max \{B_\varPhi , B_\varPsi \}. $$

In particular, this implies that unions, direct sums and sums of normalised tight frames are again normalised tight frames.

(b) Show that (5.13) is equivalent to

$$\begin{aligned} S^{-1}(f+g) = S_\varPhi ^{-1}(f) + S_\varPsi ^{-1}(g), \qquad \forall f+g \in \varXi . \end{aligned}$$

(5.15)

For the choices (5.14), this gives

$$ (\varPhi \cup \varPsi )\,\tilde{} = \tilde{\varPhi }\cup \tilde{\varPsi }, \qquad (\varPhi \oplus \varPsi )\,\tilde{} = \tilde{\varPhi }\oplus \tilde{\varPsi }, \qquad (\varPhi \mathbin {\hat{+}}\varPsi )\,\tilde{} = \tilde{\varPhi }\mathbin {\hat{+}}\tilde{\varPsi }. $$

5.5.

Show that a finite frame $(f_j)_{j\in J}$ is a simple lift if and only if

$$ \sum _j f_j\ne 0, \qquad \langle \sum _j f_j, f_k\rangle =C,\quad \forall k. $$