Math Problems Vault

Problem 10.22 from Heil C. "A Basis Theory Primer" [Hei11]More problems in Chapter 10 "Sampling, Weighted Exponentials, and Translations"

Let $w(x) = \max \left\{ 1-\left|x\right|, 0\right\}$ be the "hat" or "tent" function on $[-1,1]$ . Use Lemma 10.20 to show that

\Phi _{w}(\xi ) = \frac{2+\cos 2 \pi \xi }{3} = \frac{1+2 \cos ^{2} \pi \xi }{3},

and conclude that $\mathcal{T}(w)$ is a Riesz basis for its closed span $V_{0}(g)$ . Find $\widehat{\widetilde{w}}, \Phi_{\widetilde{w}}$ , $\widehat{w^{\sharp }}$ , and $\left[\widehat{w^{\sharp }}, \widehat{\widetilde{w}}\right]$ .

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Problem 12.14 from Heil C. "A Basis Theory Primer" [Hei11]More problems in Chapter 12 "Wavelet Bases and Frames"

The hat function on $[0,2]$ is $B_{1}(x) = \max \left\{ 1-\left|x-1\right|, 0\right\}$ . Show directly that $B_{1}$ is refinable.

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Problem 12.20 from Heil C. "A Basis Theory Primer" [Hei11]More problems in Chapter 12 "Wavelet Bases and Frames"

Set $B_{0} = \chi_{[0,1]}$ , and recursively define the $n$ th $B$ -spline $B_{n}$ by

B_{n} = B_{n-1} * \chi _{[0,1]}

where * denotes the convolution operation defined in Exercise 12.19.

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(a)

Show that $B_{1} = \chi_{[0,1]} * \chi_{[0,1]}$ is the hat function on the interval $[0,2]$ , and find explicit formulas for $B_{2}$ and $B_{2}^{\prime }$ .

(b)

Show that $B_{n}$ is refinable, integrable, and compactly supported. Show that the symbol for this refinement equation is

m_{0}(\xi ) = \left(\frac{1+e^{-2 \pi i \xi }}{2}\right)^{n+1}

(c)

Find an explicit formula for $\widehat{B_{n}}$ , and show that $\widehat{B_{n}} \in L^{1}(\mathbb {R})$ for all $n > 0$ .

(d)

Prove that $B_{n}^{\prime }(x) = B_{n-1}(x-1)-B_{n-1}(x)$ for $n > 1$ .

(e)

Show that $B_{n} \in C^{n-1}(\mathbb {R})$ for $n > 0$ , and $B_{n}^{(n-1)}$ is piecewise linear.

Problem 12.28 from Heil C. "A Basis Theory Primer" [Hei11]More problems in Chapter 12 "Wavelet Bases and Frames"

(a)(b)

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(a)

Show that if $\sum_{k \in \mathbb {Z}}\left|k^{j} c_{k}\right| < \infty$ , then $m_{0}$ is $j$ -times differentiable and its $j$ th derivative is the continuous function

m_{0}^{(j)}(\xi ) = \frac{(-2 \pi i)^{j}}{2} \sum _{k \in \mathbb {Z}} k^{j} c_{k} e^{-2 \pi i k x}

Conclude that

m_{0}^{(j)}(1 / 2) = 0 \quad \Longleftrightarrow \quad \sum _{k \in \mathbb {Z}}(-1)^{k} k^{j} c_{k} = 0

and compare this to the accuracy conditions given in equation (12.46).

(b)

Let $B_{n}$ be the $n$ th $B$ -spline defined in Exercise 12.20. Show that the accuracy condition $m_{0}^{(j)}(1 / 2) = 0$ holds for $j = 0, \ldots , n$ .

B-splines

References