Shift-invariant spaces and systems of translates

Given a sequence $c = \left(c_{k}\right)_{k \in \mathbb {Z}} \in \ell^{2}(\mathbb {Z})$, let $T_{n} c$ denote the translated sequence $T_{n} c = \left(c_{k-n}\right)_{k \in \mathbb {Z}}$. Characterize those sequences $c$ such that $\left\{ T_{n} c\right\}_{n \in \mathbb {Z}}$ is complete, Bessel, minimal, a frame, a Schauder basis, a Riesz basis, or an orthonormal basis for $\ell^{2}(\mathbb {Z})$.

Let $g \in L^{2}(\mathbb {R})$ and $a > 0$ be given, and define $h(x) = g(a x)$. Show that $\left\{ g(x-a k)\right\}_{k \in \mathbb {Z}}$ is a frame sequence in $L^{2}(\mathbb {R})$ if and only if $\left\{ h(x-k)\right\}_{k \in \mathbb {Z}}$ is a frame sequence in $L^{2}(\mathbb {R})$. What is the relation between the closed spans of these two systems?

Given a function $f \in L^{1}(\mathbb {R})$ and given $a > 0$, we call the function

the $a$-periodization of $f$ (or simply the periodization if $a = 1$). Show that the series defining $\varphi$ converges absolutely in $L^{1}[0, a]$, and

In particular, the bracket product of $f, g \in L^{2}(\mathbb {R})$ is the function $[f, g] \in L^{1}(\mathbf{T})$ defined by

Show that if $g \in L^{2}(\mathbb {R})$ is not the zero function, then $\left\{ T_{a} g\right\}_{a \in \mathbb {R}}$ is finitely linearly independent.

Suppose that $g \in L^{2}(\mathbb {R})$ is such that $\mathcal{M}(\widehat{g}) = \left\{ e^{-2 \pi i k \xi } \widehat{g}(\xi )\right\}_{k \in \mathbb {Z}}$ is a Bessel sequence in $L^{2}(\mathbb {R})$. By Theorem 7.2, if $\left(c_{k}\right)_{k \in \mathbb {Z}} \in \ell^{2}(\mathbb {Z})$ then the series $F(\xi ) = \sum_{k \in \mathbb {Z}} c_{k} e^{-2 \pi i k \xi } \widehat{g}(\xi )$ converges unconditionally in $L^{2}(\mathbb {R})$. Also, since $\left\{ e^{-2 \pi i k \xi }\right\}_{k \in \mathbb {Z}}$ is an orthonormal basis for $L^{2}(\mathbf{T})$ the series $\hat{c}(\xi ) = \sum_{k \in \mathbb {Z}} c_{k} e^{-2 \pi i k \xi }$ converges unconditionally in $L^{2}(\mathbf{T})$. Show that $F(\xi ) = \widehat{c}(\xi ) \widehat{g}(\xi )$ a.e.

Prove the remaining statements in Theorem 10.19.

Suppose that $g \in L^{2}(\mathbb {R})$ is such that $\mathcal{T}(g)$ is a frame sequence in $L^{2}(\mathbb {R})$, and let $S: V_{0}(g) \rightarrow V_{0}(g)$ be its frame operator.

Suppose that $f, g \in L^{2}(\mathbb {R})$ are such that $\mathcal{T}(f)$ and $\mathcal{T}(g)$ are both orthonormal sequences.

Let $g \in L^{2}(\mathbb {R})$ be given.

Suppose that $g \in L^{2}(\mathbb {R})$ is such that $\mathcal{T}(g)$ is a frame sequence in $L^{2}(\mathbb {R})$, and let $g^{\sharp } \in L^{2}(\mathbb {R})$ be the function whose Fourier transform is

Show that $g^{\sharp } \in V_{0}(g) = \overline{\operatorname {span}}(\mathcal{T}(g))$, and $\mathcal{T}\left(g^{\sharp }\right)$ is a Parseval frame for $V_{0}(g)$. Show further that if $\mathcal{T}(g)$ is a Riesz basis for $V_{0}(g)$ then $\mathcal{T}\left(g^{\sharp }\right)$ is an orthonormal basis for $V_{0}(g)$.

Let $w(x) = \max \left\{ 1-\left|x\right|, 0\right\}$ be the "hat" or "tent" function on $[-1,1]$. Use Lemma 10.20 to show that

and conclude that $\mathcal{T}(w)$ is a Riesz basis for its closed span $V_{0}(g)$. Find $\widehat{\widetilde{w}}, \Phi_{\widetilde{w}}$, $\widehat{w^{\sharp }}$, and $\left[\widehat{w^{\sharp }}, \widehat{\widetilde{w}}\right]$.

Suppose that $g \in L^{2}(\mathbb {R})$ and $a_{k} \in \mathbb {R}$ are such that $\left\{ g\left(x-a_{k}\right)\right\}_{k \in \mathbb {N}}$ is a Schauder basis for $L^{2}(\mathbb {R})$. Show that the sequence $\left(a_{k}\right)_{k \in \mathbb {N}}$ must be uniformly separated, i.e., $\inf_{j \neq k}\left|a_{j}-a_{k}\right| > 0$.

Given $g \in L^{2}(\mathbb {R})$, show that $\mathcal{T}(g) = \left\{ T_{k} g\right\}_{k \in \mathbb {Z}}$ is a Riesz basis for its closed span with frame bounds $A, B$ if and only if equation (11.8) holds.