Math Problems Vault

Problem 12.10 from Heil C. "A Basis Theory Primer" [Hei11]More problems in Chapter 12 "Wavelet Bases and Frames"

Prove that the Shannon MRA $\left\{ V_{n}\right\}_{n \in \mathbb {Z}}$ constructed in Example 12.12 is indeed an MRA.

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Problem 12.11 from Heil C. "A Basis Theory Primer" [Hei11]More problems in Chapter 12 "Wavelet Bases and Frames"

Prove Lemma 12.10.

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Remark.

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Lemma 12.10: Suppose that $\left\{ V_{n}\right\}_{n \in \mathbb {Z}}$ is an MRA for $L^{2}(\mathbb {R})$ and let $P_{n}$ denote the orthogonal projection of $L^{2}(\mathbb {R})$ onto $V_{n}$ . Then the following statements hold. (a) $V_{n} = D_{2^{n}}(V_{0}) = \left\{ f(2^{n} x) : f \in V_{n}\right\}$ . (b) $\left\{ D_{2^{n}} T_{k} \varphi \right\}_{k \in \mathbb {Z}}$ is an orthonormal basis for $V_{n}$ . (c) $V_{0}$ is shift-invariant, and $V_{n}$ is $2^{-n}$ -shift-invariant. (d) $P_{n} f \rightarrow f$ in $L^{2}(\mathbb {R})$ as $n \rightarrow \infty$ for every $f \in L^{2}(\mathbb {R})$ . (e) $P_{n} f \rightarrow 0$ in $L^{2}(\mathbb {R})$ as $n \rightarrow -\infty$ for every $f \in L^{2}(\mathbb {R})$ .

Problem 12.12 from Heil C. "A Basis Theory Primer" [Hei11]More problems in Chapter 12 "Wavelet Bases and Frames"

(a)(b)

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(a)

Suppose that $\left\{ V_{n}\right\}_{n \in \mathbb {Z}}$ is an MRA for $L^{2}(\mathbb {R})$ with scaling function $\varphi$ . Show that $\left\{ D_{2^{n}} T_{k} \varphi \right\}_{k, n \in \mathbb {Z}}$ is complete in $L^{2}(\mathbb {R})$ .

(b)

Show that $\left\{ D_{2^{n}} T_{k} \chi_{[0,1]}\right\}_{k, n \in \mathbb {Z}}$ is complete in $L^{2}(\mathbb {R})$ .

Problem 12.22 from Heil C. "A Basis Theory Primer" [Hei11]More problems in Chapter 12 "Wavelet Bases and Frames"

Suppose $\varphi \in L^{2}(\mathbb {R})$ is refinable with refinement coefficients $\left(c_{k}\right)_{k \in \mathbb {Z}} \in \ell^{2}(\mathbb {Z})$ . Show that if $\varphi$ is compactly supported and $\left\{ T_{k} \varphi \right\}_{k \in \mathbb {Z}}$ is orthonormal, then only finitely many of the refinement coefficients are nonzero.

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Problem 12.23 from Heil C. "A Basis Theory Primer" [Hei11]More problems in Chapter 12 "Wavelet Bases and Frames"

Show that equations (12.42) and (12.43) are equivalent when $\left(c_{k}\right)_{k \in \mathbb {Z}} \in \ell^{1}(\mathbb {Z})$ .

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Problem 12.37 from Heil C. "A Basis Theory Primer" [Hei11]More problems in Chapter 12 "Wavelet Bases and Frames"

(a)(b)(c)

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(a)

Prove statement (a) in Theorem 12.33.

(b)

Prove statement (b) in Theorem 12.33.

(c)

Justify equations (12.63) and (12.64) in the proof of statement (c) of that theorem.

Problem 12.38 from Heil C. "A Basis Theory Primer" [Hei11]More problems in Chapter 12 "Wavelet Bases and Frames"

(a)(b)(c)

Suppose that $\varphi \in L^{2}(\mathbb {R})$ is the scaling function for an MRA, and the wavelet $\psi$ is defined as in Theorem 12.33. Prove the following statements.

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(a)

$\left|\widehat{\varphi }(\xi )\right|^{2}+\left|\widehat{\psi }(\xi )\right|^{2} = \left|\widehat{\varphi }(\xi / 2)\right|^{2}$ a.e.

(b)

$\sum_{n = 1}^{\infty }\left|\widehat{\psi }\left(2^{n} \xi \right)\right|^{2} = \left|\widehat{\varphi }(\xi )\right|^{2}$ a.e.

(c)

$\sum_{n \in \mathbb {Z}}\left|\widehat{\psi }\left(2^{n} \xi \right)\right|^{2} = 1$ a.e. (try to do this without appealing to Theorem 12.6).

Remark This problem is easier if we assume that $\varphi \in L^{1}(\mathbb {R}) \cap L^{2}(\mathbb {R})$ , for then $\widehat{\varphi } \in C_{0}(\mathbb {R})$ by Theorem 9.10.

Problem 12.39 from Heil C. "A Basis Theory Primer" [Hei11]More problems in Chapter 12 "Wavelet Bases and Frames"

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

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(a)

Show that the Journé set $E$ defined in equation (12.65) is a wavelet set (use Exercise 12.4).

(b)

Define $\widehat{\psi } = \chi_{E}$ , and suppose that $\psi$ was associated with an MRA. Let $\varphi$ be the scaling function for this MRA. Use Exercise 12.38 to show that $\left|\widehat{\varphi }\right| = \chi_{F}$ , where

F = \left[-\frac{8}{7},-1\right] \cup \left[-\frac{4}{7},-\frac{1}{2}\right] \cup \left[-\frac{2}{7}, \frac{2}{7}\right] \cup \left[\frac{1}{2}, \frac{4}{7}\right] \cup \left[1, \frac{8}{7}\right].

(c)

Use the equation $\widehat{\varphi }(\xi ) = m_{0}(\xi / 2) \widehat{\varphi }(\xi / 2)$ to show that $\left|m_{0}(\xi )\right| = 1$ for $\xi \in \left[-\frac{2}{7}, \frac{2}{7}\right]$ . Since $m_{0}$ is 1 -periodic, this also shows that $\left|m_{0}(\xi )\right| = 1$ for $\xi \in \left[\frac{5}{7}, \frac{9}{7}\right]$ . Show that if $\xi \in \left[1, \frac{8}{7}\right]$ then $\varphi (2 \xi ) = 0$ but $\left|m_{0}(\xi ) \varphi (\xi )\right| = 1$ . Conclude that $\psi$ cannot be associated with an MRA.

(d)

For an alternative proof that $\psi$ cannot be associated with an MRA, compute $\Phi_{\varphi }(\xi ) = \sum_{k \in \mathbb {Z}}\left|\widehat{\varphi }(\xi +k)\right|^{2}$ , and use this to obtain a contradiction.

Problem 12.40 from Heil C. "A Basis Theory Primer" [Hei11]More problems in Chapter 12 "Wavelet Bases and Frames"

(a)(b)

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(a)

Define

\nu (x) = \begin{cases} 0, & x < 0 \\ x^{4}\left(35-84 x+70 x^{2}-20 x^{3}\right), & 0 \leq x \leq 1 \\ 1, & x > 1\end{cases}

Show that $\nu \in C^{3}(\mathbb {R})$ and $0 \leq \nu (x) \leq 1$ for all $x$ . We call $\nu$ a $C^{3}$ -sigmoid function.

(b)

Define

\widehat{\varphi }(\xi ) = \begin{cases} 0, & \xi \leq -\frac{2}{3} \text{ or } \xi \geq \frac{2}{3} \\ \sin \left(\frac{\pi }{2} \nu (3 x+2)\right), & -\frac{2}{3} < \xi < -\frac{1}{3} \\ 1, & -\frac{1}{3} \leq \xi \leq \frac{1}{3} \\ \cos \left(\frac{\pi }{2} \nu (3 x+2)\right), & \frac{1}{3} < \xi < \frac{2}{3}\end{cases}

Show that $\widehat{\varphi }$ is a bell function in the sense of Example 12.39.

Multiresolution analysis

References