Gabor Bases and Frames

Use the fact that $T_{a} M_{b} = e^{-2 \pi i a b} M_{b} T_{a}$ to show that the set $\left\{ T_{a} M_{b}\right\}_{a, b \in \mathbb {R}}$ of time-frequency shift operators is not closed under compositions, and hence does not form a group.

Define

and show that $\mathbb {H}_{1}$ is a nonabelian group under composition of operators.

Define

Show that $\mathbb {H}_{2}$ is a nonabelian group with respect to the operation

Show further that $\mathbb {H}_{2}$ is isomorphic to $\mathbb {H}_{1}$.

Define

Show that $\mathbb {H}_{3}$ is a nonabelian group with respect to multiplication of matrices, and $\mathbb {H}_{3}$ is isomorphic to $\mathbb {H}_{1}$.

Show that $a \mathbb {Z} \times b \mathbb {Z} \times \left\{ 0\right\} = \left\{ (a k, b n, 0)\right\}_{k, n \in \mathbb {Z}}$ is not a subgroup of $\mathbb {H}_{2}$, but the countable subset $a \mathbb {Z} \times b \mathbb {Z} \times a b \mathbb {Z} = \left\{ (a k, b n, a b j)\right\}_{k, n, j \in \mathbb {Z}}$ is a subgroup.

As a set, $\mathbb {H}_{2} = \mathbb {R}^{3}$, and hence has a natural topology. In fact, $\mathbb {H}_{2}$ is an example of a locally compact group (LCG). Every LCG has associated left and right Haar measures (and these are unique up to scalar multiples). Show that the left Haar measure for $\mathbb {H}_{2}$ is $d a d b d t$, which means that for every $(c, d, u) \in \mathbb {H}_{2}$ we have

for every integrable function $F$ on $\mathbb {H}_{2} = \mathbb {R}^{3}$. Show that the right Haar measure is also $d a d b d t$. Thus, even though $\mathbb {H}_{2}$ is nonabelian, its left and Haar right measures coincide (such an LCG is said to be unimodular).

Show that the frame operator $S$ commutes with $M_{b n}$ and $T_{a k}$ for all $k, n \in \mathbb {Z}$, and use this to show that $S^{-1}$ also commutes with $M_{b n}$ and $T_{a k}$.

Show that the canonical dual frame of $\mathcal{G}(g, a, b)$ is the Gabor frame $\mathcal{G}(\widetilde{g}, a, b)$ where $\widetilde{g} = S^{-1} g$.

Suppose that $\mathcal{G}(g, a, b)$ is a frame for $L^{2}(\mathbb {R})$. Show that $\mathcal{G}(g, a, b)$ is a Riesz basis if and only if $\left\langle g, \widetilde{g} \right\rangle = 1$.

Show that the canonical Parseval frame of a lattice Gabor frame is another lattice Gabor frame. Specifically, if $\mathcal{G}(g, a, b)$ is a frame for $L^{2}(\mathbb {R})$ and we set $g^{\sharp } = S^{-1 / 2} g$, where $S$ is the frame operator, then $\mathcal{G}\left(g^{\sharp }, a, b\right)$ is a Parseval frame for $L^{2}(\mathbb {R})$.

Show that $\mathcal{G}\left(\chi_{[0,1]}, 1,1\right)$ is an orthonormal basis for $L^{2}(\mathbb {R})$.

Show that $\mathcal{G}\left(\chi_{[0,1]}, a, 1\right)$ is a frame for $L^{2}(\mathbb {R})$ if and only if $0 < a \leq 1$.

The frame operator is pointwise multiplication by $b^{-1} G_{0}$, i.e., $S f = b^{-1} G_{0} f$ for $f \in L^{2}(\mathbb {R})$.

The canonical dual frame is $\mathcal{G}(\widetilde{g}, a, b)$ where $\widetilde{g} = b g / G_{0}$.

If $a b = 1$ then $\mathcal{G}(g, a, b)$ is a Riesz basis for $L^{2}(\mathbb {R})$.

If $0 < a b < 1$ then $\mathcal{G}(g, a, b)$ is a redundant frame for $L^{2}(\mathbb {R})$.

Show that for every $n \in \mathbb {N}$, there exists a polynomial $p_{n}$ of degree $3 n$ such that

Conclude that $f$ is infinitely differentiable, every derivative of $f$ is bounded, and $f^{(n)}(x) = 0$ for every $x \leq 0$ and $n \geq 0$.

Show that if $a < b$, then $g(x) = f(x-a) f(b-x)$ is infinitely differentiable, is zero outside of $(a, b)$, and is strictly positive on $(a, b)$.

Set $F_{0}(x) = \sum_{k \in \mathbb {Z}}\left|f(x-a k)\right|^{2}$ and show that $g = f / F_{0}^{1 / 2}$ is infinitely differentiable, compactly supported, and satisfies $\sum_{k \in \mathbb {Z}}\left|g(x-a k)\right|^{2} = 1$ everywhere.

Show that there exists a function $g \in C_{c}^{\infty }(\mathbb {R})$ such that $\mathcal{G}(g, a, b)$ is a Parseval frame for $L^{2}(\mathbb {R})$.

Show that $\mathcal{G}(g, a, b)$ is a Riesz basis for $L^{2}(\mathbb {R})$ if and only if it is a frame and $a b = 1$.

Show that $\mathcal{G}(g, a, b)$ is an orthonormal basis for $L^{2}(\mathbb {R})$ if and only if it is a tight frame, $a b = 1$, and $\left\| g\right\|_{L^{2}} = 1$.

$W\left(L^{\infty }, \ell^{1}\right) \subseteq L^{p}(\mathbb {R})$ for $1 \leq p \leq \infty$, and $W\left(L^{\infty }, \ell^{1}\right)$ is dense in $L^{p}(\mathbb {R})$ for $1 \leq p < \infty$.

$W\left(L^{\infty }, \ell^{1}\right)$ is closed under translations: $\left\| T_{b} f\right\|_{W(L^{\infty }, \ell^{1})} \leq 2\left\| f\right\|_{W(L^{\infty }, \ell^{1})}$.

$W\left(L^{\infty }, \ell^{1}\right)$ is an ideal in $L^{\infty }(\mathbb {R})$ under pointwise products: if $f \in L^{\infty }(\mathbb {R})$ and $g \in W\left(L^{\infty }, \ell^{1}\right)$, then $f g \in W\left(L^{\infty }, \ell^{1}\right)$ and $\left\| f g\right\|_{W(L^{\infty }, \ell^{1})} \leq \left\| f\right\|_{L^{\infty }}\left\| g\right\|_{W(L^{\infty }, \ell^{1})}$.

$\displaystyle \frac{1}{C_{1/a}}\lvert \lvert \lvert f\rvert \rvert \rvert_{a} \leq \left\| f\right\|_{W(L^{\infty }, \ell^{1})}$, where $\lvert \lvert \lvert f\rvert \rvert \rvert_{a} = \sum_{k \in \mathbb {Z}}\left\| f \cdot \chi_{[a k, a(k+1)]}\right\|_{L^{\infty }}$.

Justify the equality passing from $\int f(x) e^{-2 \pi i b n x} \overline{g(x-a k)} \, d x$ to its periodized double-sum form.

Justify the equality obtained by integrating $f(x) \overline{g(x-a k)} F_{k}(x)$ periodically versus over all of $\mathbb {R}$.

Justify the interchange of the sum over $k$ with the sum over $j$ and the integral.

Justify the interchange of the sum over $j$ with the outer integral.

Let $g \in L^{2}(\mathbb {R})$ and $a, b > 0$ be such that $\mathcal{G}(g, a, b)$ is a frame for $L^{2}(\mathbb {R})$. Show that there exists a $\delta > 0$ such that if $h \in L^{2}(\mathbb {R})$ and $\left\| g-h\right\|_{W\left(L^{\infty }, \ell^{1}\right)} < \delta$, then $\mathcal{G}(h, a, b)$ is a frame for $L^{2}(\mathbb {R})$.

Does part (a) remain valid if we replace the amalgam norm $\left\| \cdot \right\|_{W\left(L^{\infty }, \ell^{1}\right)}$ by the $L^{2}$-norm $\left\| \cdot \right\|_{L^{2}}$?

$f(x) = \int_{0}^{1} Z f(x, \xi ) d \xi$ for almost every $x \in \mathbb {R}$.

If $Z f$ is continuous, then $f$ is continuous.

$Z$ is an injective mapping of $L^{1}(\mathbb {R})$ into $L^{1}(Q)$, and the range of $Z: L^{1}(\mathbb {R}) \rightarrow L^{1}(Q)$ is a proper, dense subspace of $L^{1}(Q)$.

$Z^{-1}: L^{1}(Q) \rightarrow L^{1}(\mathbb {R})$ is unbounded.

Show that if $f$ is even then $\operatorname {Zf}(1 / 2,1 / 2) = 0$.

Show that if $f$ is odd then $Z f(0,0) = Z f(0,1 / 2) = Z f(1 / 2,0) = 0$.

Show that if $f$ is real valued then $Z f(x, 1 / 2) = 0$ for some $x \in [0,1]$.

Let $p(x) = \sum_{k = -N}^{N} c_{k} e^{2 \pi i k x}$ be a trigonometric polynomial. Show that if $\left|p\right| = 1$ a.e., then $p(x) = c_{n} e^{2 \pi i n x}$ for some $n$ between $-N$ and $N$.

Suppose that $g \in L^{2}(\mathbb {R})$ is compactly supported. Show that $\mathcal{G}(g, 1,1)$ is an orthonormal basis for $L^{2}(\mathbb {R})$ if and only if $\left|g\right| = \chi_{E}$ for some bounded set $E \subseteq \mathbb {R}$ that satisfies $\sum_{k \in \mathbb {Z}} \chi_{E}(x-k) = 1$ a.e.

Show that if $A, B$ are self-adjoint in the sense that

then

where $[A, B] = A B-B A$ is the commutator of $A$ and $B$.

Show that equality holds in part (a) if and only if $A f = i c B f$ for some $c \in \mathbb {R}$.

Fix $z \in \mathbb {R}^{2}$. Show that $\mathcal{G}(g, \Lambda )$ is linearly independent if and only if $\mathcal{G}(g, \Lambda +z)$ is linearly independent.

Given $r \in \mathbb {R}$, let $S_{r} = \left[\begin{array}{ll}1 & 0 \\ r & 1\end{array}\right]$, so multiplication by the matrix $S_{r}$ is a shear operation on $\mathbb {R}^{2}$. Define $h(x) = e^{\pi i r x^{2}} g(x)$, and show that $\mathcal{G}(g, \Lambda )$ is linearly independent if and only if $\mathcal{G}\left(h, S_{r}(\Lambda )\right)$ is linearly independent.

Let $R = \left[\begin{array}{cc}0 & -1 \\ 1 & 0\end{array}\right]$, so multiplication by $R$ is a counterclockwise rotation of $\mathbb {R}^{2}$ by 90 degrees. Show that $\mathcal{G}(g, \Lambda )$ is linearly independent if and only if $\mathcal{G}(\stackrel{\vee }{g}, R(\Lambda ))$ is linearly independent, where $\stackrel{\vee }{g}$ is the inverse Fourier transform of $g$.

Given $a \neq 0$, let $D_{a} = \left[\begin{array}{cc}a & 0 \\ 0 & 1 / a\end{array}\right]$, so multiplication by $D_{a}$ is a dilation by $a$ on the $x_{1}$-axis and a corresponding dilation by $1 / a$ on the $x_{2}$-axis. Define $h(x) = g(x / a)$, and show that $\mathcal{G}(g, \Lambda )$ is linearly independent if and only if $\mathcal{G}\left(h, D_{a}(\Lambda )\right)$ is linearly independent.

Show that if $A$ is a $2 \times 2$ matrix with $\operatorname {det}(A) = 1$, then $A$ can be written as a product of matrices of the form $S_{r}, R$, and $D_{a}$.

Show that if $\left\{ q_{k}\right\}_{k = 1}^{N}$ is any set of finitely many distinct real numbers, then $\left\{ M_{q_{k}} g\right\}_{k = 1}^{N}$ is linearly independent.

Show that if $\left\{ p_{k}\right\}_{k = 1}^{N}$ is any set of finitely many distinct real numbers, then $\left\{ T_{p_{k}} g\right\}_{k = 1}^{N}$ is linearly independent.

Show that if $\Lambda = \left\{ \left(p_{k}, q_{k}\right)\right\}_{k = 1}^{N}$ is any set of finitely many distinct but collinear points in $\mathbb {R}^{2}$, then $\mathcal{G}(g, \Lambda ) = \left\{ M_{q_{k}} T_{p_{k}} g\right\}_{k = 1}^{N}$ is linearly independent.

Prove that

Use equation (11.41) to show that $H_{n}(x) = p_{n}(x) e^{-\pi x^{2}}$, where $p_{n}$ is a polynomial of degree $n$ whose leading coefficient is $(-4 \pi )^{n}$. Consequently, each $H_{n}$ is infinitely differentiable and has exponential decay at infinity, and $\operatorname {span}\left\{ H_{n}\right\}_{n \geq 0} = \left\{ p(x) e^{-\pi x^{2}}: p \text{ is a polynomial}\right\}$.