Math Problems Vault

Problem 11.1 from Heil C. "A Basis Theory Primer" [Hei11]More problems in Chapter 11 "Gabor Bases and Frames"

Given $g \in L^{2}(\mathbb {R})$ , show that $\left\{ M_{b n} T_{a k} g\right\}_{k, n \in \mathbb {Z}}$ is a frame for $L^{2}(\mathbb {R})$ if and only if $\left\{ T_{a k} M_{b n} g\right\}_{k, n \in \mathbb {Z}}$ is a frame, and in this case their frame operators coincide.

?

Problem 11.3 from Heil C. "A Basis Theory Primer" [Hei11]More problems in Chapter 11 "Gabor Bases and Frames"

(a)(b)(c)(d)

Let $\mathcal{G}(g, a, b)$ be a Gabor frame for $L^{2}(\mathbb {R})$ .

?

(a)

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 $T_{a k}$ .

(b)

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

(c)

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 $\left\langle g, \widetilde{g} \right\rangle = 1$ .

(d)

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 $L^{2}(\mathbb {R})$ .

Problem 11.4 from Heil C. "A Basis Theory Primer" [Hei11]More problems in Chapter 11 "Gabor Bases and Frames"

Fix $g \in L^{2}(\mathbb {R})$ and $a, b > 0$ . Recall from equations (9.2) and (9.3) that the Fourier transform interchanges translation with modulation. Use this to show that

\mathcal{G}(g, a, b) \text{ is a frame } \Longleftrightarrow \mathcal{G}(\widehat{g}, b, a) \text{ is a frame. }

?

Problem 11.5 from Heil C. "A Basis Theory Primer" [Hei11]More problems in Chapter 11 "Gabor Bases and Frames"

(a)(b)

?

(a)

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

(b)

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

Remark Amazingly, there is no known explicit characterization of the set of points $(a, b)$ such that $\mathcal{G}\left(\chi_{[0,1]}, a, b\right)$ is a frame for $L^{2}(\mathbb {R})$ , see [Jan03].

Problem 11.6 from Heil C. "A Basis Theory Primer" [Hei11]More problems in Chapter 11 "Gabor Bases and Frames"

(a)(b)(c)(d)

Assume that the hypotheses of part (a) of Theorem 11.4 are satisfied, i.e., $0 < a b \leq 1, \operatorname {supp}\left(g\right) \subseteq \left[0, b^{-1}\right]$ , and equation (11.3) holds. Prove the following statements about the frame $\mathcal{G}(g, a, b)$ .

?

(a)

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

(b)

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

(c)

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

(d)

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

Problem 11.7 from Heil C. "A Basis Theory Primer" [Hei11]More problems in Chapter 11 "Gabor Bases and Frames"

Show that if $g \in C_{c}(\mathbb {R})$ is not the zero function, then there exist some $a, b > 0$ such that $\mathcal{G}(g, a, b)$ is a frame for $L^{2}(\mathbb {R})$ .

?

Problem 11.8 from Heil C. "A Basis Theory Primer" [Hei11]More problems in Chapter 11 "Gabor Bases and Frames"

Let $g \in C_{c}(\mathbb {R})$ satisfy $\operatorname {supp}\left(g\right) = \left[0, b_{0}^{-1}\right]$ and $g(x) > 0$ for $x \in \left(0, b_{0}^{-1}\right)$ . Show that $\mathcal{G}(g, a, b)$ is a frame for $L^{2}(\mathbb {R})$ for $0 < a < b_{0}^{-1}$ and $0 < b < b_{0}$ .

?

Problem 11.10 from Heil C. "A Basis Theory Primer" [Hei11]More problems in Chapter 11 "Gabor Bases and Frames"

(a)(b)

Let $0 < a b < 1$ be fixed. By Exercise 11.9, there exists a function $f \in C_{c}^{\infty }(\mathbb {R})$ supported in $\left[0, b^{-1}\right]$ such that $f > 0$ on $\left(0, b^{-1}\right)$ .

?

(a)

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.

(b)

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 $L^{2}(\mathbb {R})$ .

Problem 11.11 from Heil C. "A Basis Theory Primer" [Hei11]More problems in Chapter 11 "Gabor Bases and Frames"

(a)(b)

Fix $g \in L^{2}(\mathbb {R})$ and $a, b > 0$ .

?

(a)

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 $a b = 1$ .

(b)

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 $\left\| g\right\|_{L^{2}} = 1$ .

Problem 11.13 from Heil C. "A Basis Theory Primer" [Hei11]More problems in Chapter 11 "Gabor Bases and Frames"

Suppose that $\mathcal{G}(g, a, b)$ is a frame for $L^{2}(\mathbb {R})$ . Without appealing to Theorem 11.9, show that $\left\{ M_{k / a} \widehat{g}\right\}_{n \in \mathbb {Z}}$ and $\left\{ M_{n / b} \widehat{g}\right\}_{n \in \mathbb {Z}}$ are Riesz sequences in $L^{2}(\mathbb {R})$ .

?

Problem 11.14 from Heil C. "A Basis Theory Primer" [Hei11]More problems in Chapter 11 "Gabor Bases and Frames"

Assuming Theorem 11.9, show that $\mathcal{G}(g, a, b)$ is a tight frame for $L^{2}(\mathbb {R})$ if and only if $\mathcal{G}(g, 1 / b, 1 / a)$ is an orthogonal sequence in $L^{2}(\mathbb {R})$ .

?

Problem 11.16 from Heil C. "A Basis Theory Primer" [Hei11]More problems in Chapter 11 "Gabor Bases and Frames"

Let $g$ be the function defined in Example 11.11. Show that $\mathcal{G}(g, 1,1)$ is an orthonormal basis for $L^{2}(\mathbb {R})$ , but $g \notin W\left(L^{\infty }, \ell^{1}\right)$ .

?

Problem 11.18 from Heil C. "A Basis Theory Primer" [Hei11]More problems in Chapter 11 "Gabor Bases and Frames"

(a)(b)(c)(d)

Justify the use of Fubini's Theorem in the proof of Theorem 11.18.

?

(a)

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.

(b)

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

(c)

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

(d)

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

Problem 11.19 from Heil C. "A Basis Theory Primer" [Hei11]More problems in Chapter 11 "Gabor Bases and Frames"

(a)(b)

This exercise gives a perturbation theorem for Gabor frames.

?

(a)

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 $L^{2}(\mathbb {R})$ .

(b)

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 $\left\| \cdot \right\|_{L^{2}}$ ?

Problem 11.24 from Heil C. "A Basis Theory Primer" [Hei11]More problems in Chapter 11 "Gabor Bases and Frames"

Prove the remaining statements in Theorem 11.31.

?

Remark.

?

Theorem 11.31: Let $g \in L^{2}(\mathbb {R})$ be fixed. (a) $\mathcal{G}(g, 1, 1)$ is complete in $L^{2}(\mathbb {R})$ if and only if $Z g \neq 0$ a.e. (b) $\mathcal{G}(g, 1, 1)$ is minimal in $L^{2}(\mathbb {R})$ if and only if $1 / Z g \in L^{2}(Q)$ ; in this case $\mathcal{G}(g, 1, 1)$ is exact and its biorthogonal system is $\mathcal{G}(\widetilde{g}, 1, 1)$ where $\widetilde{g} \in L^{2}(\mathbb {R})$ satisfies $Z\widetilde{g} = 1 / Z g$ . (c) $\mathcal{G}(g, 1, 1)$ is a Bessel sequence in $L^{2}(\mathbb {R})$ if and only if $Z g \in L^{\infty }(\mathbb {R})$ , and in this case $B = \left\| Z g\right\|_{L^{\infty }}^{2}$ is a Bessel bound. (d) $\mathcal{G}(g, 1, 1)$ is a frame for $L^{2}(\mathbb {R})$ if and only if there exist $A, B > 0$ such that $A \leq \left|Z g(x, \xi )\right|^{2} \leq B$ a.e.; in this case $\mathcal{G}(g, 1, 1)$ is a Riesz basis and $A, B$ are frame bounds. (e) $\mathcal{G}(g, 1, 1)$ is an orthonormal basis for $L^{2}(\mathbb {R})$ if and only if $\left|Z g(x, \xi )\right| = 1$ a.e.

Problem 11.25 from Heil C. "A Basis Theory Primer" [Hei11]More problems in Chapter 11 "Gabor Bases and Frames"

(a)(b)

?

(a)

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 $N$ .

(b)

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.

Problem 11.26 from Heil C. "A Basis Theory Primer" [Hei11]More problems in Chapter 11 "Gabor Bases and Frames"

Let $\phi (x) = e^{-\pi x^{2}}$ be the Gaussian function. Show that $\mathcal{G}(\phi , 1,1)$ is $\ell^{2}$ -independent, i.e., if $c = \left(c_{k n}\right)_{k, n \in \mathbb {Z}} \in \ell^{2}\left(\mathbb {Z}^{2}\right)$ and $\sum c_{k n} M_{n} T_{k} \phi = 0$ , then $c_{k n} = 0$ for every $k$ and $n$ . Note that since $\mathcal{G}(\phi , 1,1)$ is a Bessel sequence, if we let $R$ denote the synthesis operator for $\mathcal{G}(\phi , 1,1)$ then $\ell^{2}$ -independence is equivalent to the statement that $R: \ell^{2}\left(\mathbb {Z}^{2}\right) \rightarrow L^{2}(\mathbb {R})$ is injective.

?

Gabor systems and frames

References