Central Approximation Space

First of all, construct a central approximation space $\mathbf{V}_0$ , which is a subspace of the space of square-integrable functions $L^2(-\infty ,\infty )$ , which (a) is spanned by a translation-generated (a.k.a. ``Riesz'') basis

$$\mathbf{V}_0=span\{\phi(t-l):~\ell=0,\pm 1,\cdots\}$$

and (b) is orthonormal:

$$\int_{-\infty}^\infty \overline\phi(t-k)\phi(t-\ell)\, dt=\delta_{k\ell}~.$$ (286)

The existence of such a basis is equivalent to the statement that $\mathbf{V}_0$ is closed under integral shifts of its elements, i.e.

$$f(t)\in\mathbf{V}_0\Rightarrow f(t-\ell)\in\mathbf{V}_0~\textrm{whenever}~ \ell=integer~.$$

The function $\phi(t)$ , known as a scaling function (a.k.a. ``father wavelet'') can be any square integrable function as long as it satisfies the integer-shifted orthonormality condition, Eq.(2.86).

A particular example of such a basis is the set of wave packets $\{ Q^\varepsilon_{0\ell}(t)\}$ , Eq.(2.72) on page :

$$Q^\varepsilon_{0\ell}(t) = \sqrt{\frac{1}{2\pi\varepsilon}} \int^{\varepsilon/2}_{-\varepsilon/2} e^{-2\pi i\ell \omega/\varepsilon}~e^{i\omega t} d\omega$$ (287)

$$\equiv \phi\left(t-\frac{2\pi\ell}{\varepsilon}\right)$$

For this basis the scaling function is obtained by setting $\varepsilon=2\pi$ and letting $\ell =0$ :

$$\phi(t)=\frac{\sin \pi t}{\pi t}~.$$ (288)

This scaling function happens to be one whose Fourier transform has compact support and is piecewise constant:

$$\hat\phi (\omega)=\left\{ \begin{array}{cc} \displaystyle\sqrt{\frac{1}{2\pi}}&\omega\in [-\pi,\pi]\\ 0 & \textrm{otherwise} \end{array}\right.~.$$

The central approximation space $\mathbf{V}_0$ is spanned by the orthonormal basis

$$\phi(t-\ell)=\frac{\sin \pi (t-\ell)}{\pi (t-\ell)}\quad \ell=0,\pm1,\cdots~.$$ (289)

It is the vector space of ``band-limited'' functions, i.e. functions whose Fourier transforms have compact support on the frequency interval $[-\pi , \pi ]$ . The basis for this space is generated by Eq.(2.88) and it is called the Shannon basis.

Figure 2.23: Partitioning of phase space by a collection hierarchical sets of band limited orthonormal basis functions. The heavy-lined rectangles are the phase space cells of low resolution wave packets; they span the $k=1$ st resolution vector space $\mathbf V_1$ . The shaded rectangles are those of the next (i.e. more refined) resolution wave packets; they span the $k=0$ th resolution vector space $\mathbf V_0$ . The thin and tall unshaded rectangles are those of the wave packets of still higher resolution. They span the $k=-1$ st resolution vector space $\mathbf V_{-1}$ . The unshaded rectangle in the middle is the phase space cell of the ``father wavelet'', the scaling function in Eq.(2.88). It yields (by compression and translation) all basis functions for all the vector spaces $\mathbf V_k$ .