Refinement equations and refinable functions

$\left\{ T_{k} \varphi \right\}_{k \in \mathbb {Z}}$ is a Bessel sequence, or

$\left(c_{k}\right)_{k \in \mathbb {Z}} \in \ell^{1}(\mathbb {Z})$, then $\varphi$ is refinable.

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 $B_{2}^{\prime }$.

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

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

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

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

$\operatorname {supp}\left(\varphi \right) \subseteq [0, N]$, i.e., $\varphi (x) = 0$ for almost every $x \notin [0, N]$.

If $\varphi$ is continuous then the vector $(\varphi (1), \ldots , \varphi (N-1))$ is an eigenvector of the $(N-1) \times (N-1)$ matrix

for the eigenvalue 1. Note the double-shift in the rows of $M$; for this reason $M$ is called a two-slanted matrix.

If $\varphi$ is differentiable then $\varphi^{\prime }$ is refinable.

If $\varphi \in C^{1}(\mathbb {R})$ then 1 and $1 / 2$ are both eigenvalues of $M$.

$\varphi$ cannot be infinitely differentiable, and in fact can have at most $N-2$ continuous derivatives.

Show that $\widehat{\chi }(\xi ) = e^{\pi i \xi } \frac{\sin \pi \xi }{\pi \xi }$.

Show that $\left|m_{0}(\xi )\right|^{2}+\left|m_{0}\left(\xi +\frac{1}{2}\right)\right|^{2} = 1$.

Give a direct proof of Viète's formula:

Explain how Viète's formula relates to Theorem 12.29.

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

Conclude that

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

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 $j = 0, \ldots , n$.

Show that the Hilbert transform is a unitary mapping of $L^{2}(\mathbb {R})$ onto itself, but it does not map $L^{1}(\mathbb {R}) \cap L^{2}(\mathbb {R})$ onto itself. In particular, if $f \in L^{1}(\mathbb {R}) \cap L^{2}(\mathbb {R})$ satisfies $\widehat{f}(0) \neq 0$, then $H f \notin L^{1}(\mathbb {R})$.

Suppose that $\varphi \in L^{2}(\mathbb {R})$ satisfies a refinement equation with coefficients $\left(c_{k}\right)_{k \in \mathbb {Z}} \in \ell^{1}(\mathbb {Z})$. Show that $H \varphi$ satisfies the same refinement equation.

Show that the function $1 / x$ satisfies the one-term refinement equation $\varphi (x) = 2 \varphi (2 x)$. However, $1 / x$ is not an integrable function and does not determine a tempered distribution.

The $\delta$ distribution is the linear functional on $C_{c}^{\infty }(\mathbb {R})$ defined by $\left\langle f, \delta \right\rangle = f(0)$ for $f \in C_{c}^{\infty }(\mathbb {R})$. Show that $\delta$ satisfies the equation $\varphi (x) = 2 \varphi (2 x)$ in a distributional sense.

The $j$ th distributional derivative $\delta^{(j)}$ of $\delta$ is defined by $\left\langle f, \delta^{(j)} \right\rangle = (-1)^{j} f^{(j)}(0)$ for $f \in C_{c}^{\infty }(\mathbb {R})$. Show that $\delta^{(j)}$ satisfies $\varphi (x) = 2^{j+1} \varphi (2 x)$ in a distributional sense.

The principal value of $1 / x$ is the linear functional $\operatorname {pv}(1 / x)$ on $C_{c}^{\infty }(\mathbb {R})$ given by

Show that $\operatorname {pv}(1 / x)$ satisfies $\varphi (x) = 2 \varphi (2 x)$ in a distributional sense.

Show that if the hypotheses of Theorem 12.25 are satisfied, then $\mu = \stackrel{\vee }{P}$ (the inverse distributional Fourier transform of $P$) is a tempered distribution that satisfies the refinement equation in the sense of distributions.

Show that, up to scale, $\mu$ is the unique tempered distribution that satisfies the refinement equation and has a Fourier transform that is a continuous function.

Let $\theta$ be any bounded function such that $\theta (2 \xi ) = \theta (\xi )$ for all $\xi$. Show that $\widehat{\nu }(\xi ) = P(\xi ) \theta (\xi )$ defines a tempered distribution that satisfies the refinement equation in the sense of distributions.

Show that a function $\theta$ that satisfies $\theta (2 \xi ) = \theta (\xi )$ and is continuous at $\xi = 0$ is a constant function. Give examples of nonconstant functions $\theta$ such that $\theta (2 \xi ) = \theta (\xi )$ for all $\xi$.

Let $\delta_{k}$ be the point mass at $k$ distribution, i.e., $\left\langle f, \delta_{k} \right\rangle = f(k)$ for $f \in \mathcal{S}(\mathbb {R})$. Set $\nu = \frac{1}{2} \sum_{k = 0}^{N} c_{k} \delta_{k}$, and show that the distributional Fourier transform of $\nu$ is $\widehat{\nu } = m_{0}$.

Set

Show that

and

By Exercise 12.35(a), there is a distributional solution $\mu$ to the refinement equation that satisfies $\widehat{\mu } = P$. By Theorem 12.25, $\widehat{\mu_{n}} = P_{n}$ converges to $P = \widehat{\mu }$ uniformly on compact sets. Show that $\mu_{n} \xrightarrow {\mathrm{w}^{*}} \varphi$, i.e.,

and use this to show that $\operatorname {supp}\left(\varphi \right) \subseteq [0, N]$.

Show that if there exists an integrable solution $\varphi$ to the refinement equation (12.47), then $\varphi$ is compactly supported and $\operatorname {supp}\left(\varphi \right) \subseteq [0, N]$.

Show that if the antialiasing condition is also satisfied, then there exists a compactly supported solution $\varphi \in L^{2}(\mathbb {R})$ to the refinement equation (which is therefore integrable by Cauchy-Bunyakovski-Schwarz).