Math Problems Vault

Problem 10.14 from Heil C. "A Basis Theory Primer" [Hei11]More problems in Chapter 10 "Sampling, Weighted Exponentials, and Translations"

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.

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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.

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Problem 11.34 from Heil C. "A Basis Theory Primer" [Hei11]More problems in Chapter 11 "Gabor Bases and Frames"

Show that if $\left(p_{i}, q_{i}\right)$ for $i = 1,2,3$ are three noncollinear points in $\mathbb {R}^{2}$ , then there exist an invertible $2 \times 2$ matrix $A$ and a point $z \in \mathbb {R}^{2}$ such that $\Lambda = A\left(\mathbb {Z}^{2}\right)+z$ contains those three points.

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Problem 11.35 from Heil C. "A Basis Theory Primer" [Hei11]More problems in Chapter 11 "Gabor Bases and Frames"

Fix $g \in L^{2}(\mathbb {R}) \backslash \left\{ 0\right\}$ , and let $\Lambda = \left\{ \left(p_{k}, q_{k}\right)\right\}_{k = 1}^{N}$ be any set of finitely many distinct points in $\mathbb {R}^{2}$ . Define $\mathcal{G}(g, \Lambda ) = \left\{ M_{q_{k}} T_{p_{k}} g\right\}_{k = 1}^{N}$ .

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(a)

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.

(b)

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.

(c)

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 $g$ .

(d)

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.

(e)

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 $D_{a}$ .

Remark This factorization is related to the fact that every $2 \times 2$ matrix with determinant 1 is a symplectic matrix. In contrast, not every $2 d \times 2 d$ matrix with determinant 1 is symplectic when $d > 1$ . As a consequence, the HRT Conjecture becomes even more intractable in higher dimensions.

Problem 11.36 from Heil C. "A Basis Theory Primer" [Hei11]More problems in Chapter 11 "Gabor Bases and Frames"

Assuming Theorem 11.36, prove Corollary 11.37.

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Problem 11.37 from Heil C. "A Basis Theory Primer" [Hei11]More problems in Chapter 11 "Gabor Bases and Frames"

(a)(b)(c)

Fix $g \in L^{2}(\mathbb {R}) \backslash \left\{ 0\right\}$ .

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(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.

(b)

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.

(c)

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.

Problem 11.38 from Heil C. "A Basis Theory Primer" [Hei11]More problems in Chapter 11 "Gabor Bases and Frames"

Suppose that $g \in L^{2}(\mathbb {R}) \backslash \left\{ 0\right\}$ is supported within some halfline, either $(-\infty , a]$ or $[a, \infty )$ where $a \in \mathbb {R}$ . Show that if $\Lambda = \left\{ \left(p_{k}, q_{k}\right)\right\}_{k = 1}^{N}$ is any set of finitely many distinct 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.

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Problem 11.40 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, and let $h(x) = p(x) e^{-\pi x^{2}}$ where $p$ is any nontrivial polynomial. Show that if $\Lambda = \left\{ \left(p_{k}, q_{k}\right)\right\}_{k = 1}^{N}$ is any set of finitely many distinct points in $\mathbb {R}^{2}$ , then $\mathcal{G}(h, \Lambda ) = \left\{ M_{q_{k}} T_{p_{k}} h\right\}_{k = 1}^{N}$ is linearly independent.

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Problem 11.41 from Heil C. "A Basis Theory Primer" [Hei11]More problems in Chapter 11 "Gabor Bases and Frames"

Assume that $g \in L^{2}(\mathbb {R})$ and $\Lambda = \left\{ \left(p_{k}, q_{k}\right)\right\}_{k = 1}^{N}$ are such that $\mathcal{G}(g, \Lambda )$ is linearly independent. Show that there exists $\varepsilon > 0$ such that $\mathcal{G}\left(g, \Lambda^{\prime }\right)$ is independent for any set $\Lambda^{\prime } = \left\{ \left(\alpha_{k}^{\prime }, \beta_{k}^{\prime }\right)\right\}_{k = 1}^{N}$ with $\left|\alpha_{k}-\alpha_{k}^{\prime }\right|,\left|\beta_{k}-\beta_{k}^{\prime }\right| < \varepsilon$ for $k = 1, \ldots , N$ .

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Linear independence of time-frequency translates (HRT conjecture)

References