Math Problems Vault

Problem 12.1 from Heil C. "A Basis Theory Primer" [Hei11]More problems in Chapter 12 "Wavelet Bases and Frames"

Show that if $\mathcal{W}(\psi , a, b)$ is a wavelet frame for $L^{2}(\mathbb {R})$ , then its frame operator $S$ commutes with $D_{a^{n}}$ for $n \in \mathbb {Z}$ . What does this imply about the structure of the canonical dual frame?

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Problem 12.2 from Heil C. "A Basis Theory Primer" [Hei11]More problems in Chapter 12 "Wavelet Bases and Frames"

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

Define $\mathbb {A}_{1} = \left\{ D_{a} T_{b}\right\}_{a > 0, b \in \mathbb {R}}, \quad \mathbb {A}_{2} = \left\{ \left[\begin{array}{cc}a & b \\ 0 & 1\end{array}\right]\right\}_{a > 0, b \in \mathbb {R}}, \quad \mathbb {A}_{3} = \left\{ (a, b)\right\}_{a > 0, b \in \mathbb {R}}$ . In particular, $\mathbb {A}_{1}$ is the set of all time-scale shift operators, and so is a subset of $\mathcal{B}\left(L^{2}(\mathbb {R})\right)$ . Define the following operations: On $\mathbb {A}_{1}$ it is composition of operators, on $\mathbb {A}_{2}$ it is matrix multiplication, and on $\mathbb {A}_{3}$ it is

(a, b) *(c, d) = (a c, b c+d).

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(a)

Show that $\mathbb {A}_{1}, \mathbb {A}_{2}, \mathbb {A}_{3}$ are isomorphic groups with respect to these operations.

(b)

Given $a > 1, b > 0$ , show that $\left\{ D_{a^{n}} T_{b k}\right\}_{k, n \in \mathbb {Z}}$ is not a subgroup of $\mathbb {A}_{1}$ .

(c)

With $a > 1$ and $b > 0$ fixed, let $G$ be the subgroup of $\mathbb {A}_{1}$ generated by $D_{a}$ and $T_{b}$ , i.e., it is the intersection of all subgroups of $\mathbb {A}_{1}$ that contain both $D_{a}$ and $T_{b}$ . Show that $G$ contains $D_{a^{n}} T_{\left(a^{m} j+k\right) b}$ for every $m, n, j, k \in \mathbb {Z}$ . As $\left\{ \left(a^{m} j+k\right) b\right\}_{m, j, k \in \mathbb {Z}}$ is dense in $\mathbb {R}$ , we conclude that there are no discrete "separated" subgroups of $\mathbb {A}_{1}$ that contain both dilations and translations.

(d)

As a set, $\mathbb {A}_{3} = (0, \infty ) \times \mathbb {R}$ , and hence has a natural topology. In fact, $\mathbb {A}_{3}$ 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 $\mathbb {A}_{3}$ is $\frac{d a}{a} d b$ , which means that for every $(u, v) \in \mathbb {A}_{3}$ we have

\int _{-\infty }^{\infty } \int _{0}^{\infty } F((u, v) *(a, b)) \frac{d a}{a} d b = \int _{-\infty }^{\infty } \int _{0}^{\infty } F(a, b) \frac{d a}{a} d b

for every function $F$ on $\mathbb {A}_{3}$ that is integrable with respect to $\frac{d a}{a} d b$ . What is the right Haar measure?

Remark The (isomorphic) groups $\mathbb {A}_{1}, \mathbb {A}_{2}, \mathbb {A}_{3}$ are called the affine group. The properties of the affine group should be contrasted with those of the Heisenberg affine group discussed in Exercise 11.2.

Problem 12.3 from Heil C. "A Basis Theory Primer" [Hei11]More problems in Chapter 12 "Wavelet Bases and Frames"

(a)(b)(c)

Fix $a > 1, b > 0, c > 0$ , and let $\psi \in L^{2}(\mathbb {R})$ be such that $\operatorname {supp}\left(\widehat{\psi }\right) \subseteq \left[c, c+b^{-1}\right]$ and $\sum_{n \in \mathbb {Z}}\left|\widehat{\psi }\left(a^{n} \xi \right)\right|^{2} = b$ for almost every $\xi \geq 0$ .

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(a)

Show that $\widehat{\mathcal{W}}(\psi , a, b)$ is a Parseval frame for $L^{2}[0, \infty )$ , and $\mathcal{W}(\psi , a, b)$ is a Parseval frame for $H_{+}^{2}(\mathbb {R}) = \left\{ f \in L^{2}(\mathbb {R}): \operatorname {supp}\left(\widehat{f}\right) \subseteq [0, \infty )\right\}$ .

(b)

Show that if $a, b$ are chosen correctly, then we can choose $\psi$ so that $\widehat{\psi }$ is continuous. Can we choose $\psi$ so that $\widehat{\psi }$ is infinitely differentiable?

(c)

Find $\psi_{1}, \psi_{2}$ with continuous Fourier transforms so that $\mathcal{W}\left(\psi_{1}\right) \cup \mathcal{W}\left(\psi_{2}\right)$ is a Parseval frame for $L^{2}(\mathbb {R})$ .

Problem 12.8 from Heil C. "A Basis Theory Primer" [Hei11]More problems in Chapter 12 "Wavelet Bases and Frames"

Show that $\left\{ \psi \in L^{2}(\mathbb {R}): \psi \text{ is admissible}\right\}$ is dense in $L^{2}(\mathbb {R})$ .

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Problem 12.9 from Heil C. "A Basis Theory Primer" [Hei11]More problems in Chapter 12 "Wavelet Bases and Frames"

Show that if $\psi \in L^{1}(\mathbb {R}) \cap L^{2}(\mathbb {R})$ is admissible, then $\widehat{\psi }(0) = 0$ .

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Continuous wavelet transform and wavelet frames

References