Resolution Analysis as a Decomposition into Subspaces

Fourth, decompose each resolution subspace $\mathbf V_{k}$ into its subsequent resolution subspace $\mathbf V_{k+1}$ and its corresponding orthogonal complement, the subspace of details $\mathbf O_{k+1}$ :

$$\mathbf V_{k}=\mathbf V_{k+1}\oplus \mathbf O_{k+1}$$ (284)

Given the fact that $\mathbf V_{k}$ is spanned by the $k$ th resolution basis, Eq.(2.83), the meaning of such a decomposition consists of exhibiting an alternative o.n. basis part of whose elements span $\mathbf V_{k+1}$ , while the remainder spans its orthogonal complement. This decomposition is achieved as follows:

Recall that every square integrable function $f(t)$ can be approximated as an optimal element in $\mathbf V_{k}$ . This optimal approximation, which in Section 1.5.4 was identified as the least squares approximation of $f(t)$ in the subspace $\mathbf V_{k}$ , is uniquely expressed in terms of any orthonormal basis. Following Eq.(1.8), and using the o.n. basis, Eq.(2.82), one has has the projection of $f(t)$ onto $\mathbf V_{k}$ :

$$P_{\mathbf V_{k}}f(t)=\sum_{\ell=-\infty}^\infty 2^{-k}\phi(2^{-k}t-\ell) \, \langle \phi(2^{-k}u-\ell),f(u)\rangle$$ (285)

This is the least squares approximation of $f(t)$ based on the subspace $\mathbf V_{k}$ , or more briefly the $\mathbf V_{k}$ -least squares approximation. The next (less refined) approximation is the projection of $f(t)$ onto the subspace $\mathbf V_{k+1}\subset\mathbf V_{k}$ :

$$P_{\mathbf V_{k+1}}f(t)=\sum_{\ell=-\infty}^\infty 2^{-(k+1)}\phi(2^{-(k+1)}t-\ell) \, \langle \phi(2^{-(k+1)}u-\ell),f(u)\rangle~.$$ (286)

Here $P_{\mathbf V_{k}}$ and $P_{\mathbf V_{k+1}}$ are the projection operators onto $\mathbf V_{k}$ and $\mathbf V_{k+1}$ respectively.

Let us compare the two o.n. bases for the two resolution spaces $\mathbf V_{k}$ and $\mathbf V_{k+1}$ . We shall presently see that they are two families of o.n. wave packets identified already on page by Eq.(2.64):

$$Q_{0\ell}^{\varepsilon}(t)= \left\{ \begin{array}{ccl} Q_{0\ell}^... ...Q_{0\ell}^{\varepsilon'}(t)&~~~&\varepsilon=\varepsilon' \end{array}\right.~~.$$

Here nad throughout the ensuing development we always let

$$\varepsilon'=2^{-k},\quad k=0,\pm 1,\pm 2,\cdots~~.$$

Using Eqs.(2.82) and (2.80) one finds that the $\mathbf V_{k}$ -family members are

$$\mathbf V_{k}:\, \sqrt{2^{-k}}\,\phi(2^{-k}t-\ell) =\sqrt{2^{-k}} \frac{1}{2\pi}\int_{-\pi}^\pi e^{i\omega(2^{-k}t-\ell)}d\omega$$

$$=\frac{1}{\sqrt{2\pi}}\frac{1}{\sqrt{2\varepsilon'}}\int^{\varepsilon'}_{-\varepsilon'} e^{-2\pi i\ell\omega/2\varepsilon'} e^{i\omega t} d\omega$$

$$=Q_{0\ell}^{2\varepsilon'}(t)~,$$ (287)

and

$$\Delta t = \frac{2\pi}{2\varepsilon'}=2^k$$

$$\Delta\omega = 2\varepsilon'=\frac{2\pi}{2^k}~.$$

By contrast, the $\mathbf V_{k+1}$ -family members, which are twice as wide in the temporal domain and twice as narrow in the frequency domain, are

$$\mathbf V_{k+1}:\, \sqrt{2^{-(k+1)}}\,\phi(2^{-(k+1)}t-\ell) =\sqrt{2^{-(k+1)}} \frac{1}{2\pi}\int_{-\pi}^\pi e^{i\omega(t2^{-(k+1)}-\ell)}d\omega$$

$$=\frac{1}{\sqrt{2\pi}}\frac{1}{\sqrt{\varepsilon'}} \int^{\vareps... ...'/2}_{-\varepsilon'/2} e^{-2\pi i\ell\omega/\varepsilon'} e^{i\omega t} d\omega$$

$$=Q_{0\ell}^{\varepsilon'}(t)$$ (288)

and

$$\Delta t = \frac{2\pi}{\varepsilon'}=2\times2^k$$

$$\Delta\omega = \varepsilon'=\frac{1}{2}\times\frac{2\pi}{2^k}~.$$

Figure: Partitioning of phase space by a nested set of band limited orthonormal basis functions. The tall thin rectangles and the shaded rectangles are the phasespace cells of the basis functions which span $\mathbf V_{k}$ and $\mathbf V_{k+1}$ respectively. These two sets of phase space cells are reproduced respectively on the l.h.s. and r.h.s. of Figure 2.22.

These two bases are represented by two overlapping arrays of phase space cells, as in Figure 2.21. The phase space cells referring to the $\mathbf V_{k}$ -basis are taller and skinnier than those referring to $\mathbf V_{k+1}$ . Furthermore, the phase space domain of the $\mathbf V_{k+1}$ -basis is a horizontal strip which is contained entirely within that of the $\mathbf V_{k}$ -basis. Consequently, the phase space domain of the $\mathbf V_{k}$ -basis gets partitioned into three mutually exclusive and jointly exhaustive domains:

• the negative ``band pass'' frequency strip $-\displaystyle\varepsilon'<\omega<-\displaystyle\frac{\varepsilon'}{2}$ ,
• the strip $-\displaystyle\frac{\varepsilon'}{2}<\omega<\frac{\varepsilon'}{2}$ generated by the $\mathbf V_{k+1}$ -basis, and
• The positive ``band pass'' frequency strip $$\frac{\varepsilon'}{2}<\omega<\varepsilon'$$ .

The mutual exclusivity of these three strips, together with the fact that their union equals the strip generated by the $\mathbf V_{k}$ -basis, implies that $\mathbf V_{k}$ is spanned by two fundamental bases. Besides the one given by Eq.(2.87),

$$\mathbf V_k=span\{Q_{0\ell}^{2\varepsilon'}(t):\, \ell=0,\pm1,\cdots\}~,$$

there also is

$$\mathbf V_k=span\!\!\!\!\!\!\!\! \{Q_{0\ell}^{\varepsilon'}(t):\, \ell=0,\pm1,\cdots\}\cup$$

$$\{P_{1\ell}^{\varepsilon'/2}(t):\, \ell=0,\pm1,\cdots\}\cup \{P_{-2\,\ell}^{\varepsilon'/2}(t):\, \ell=0,\pm1,\cdots\}~.$$

Figure: Two alternative partitionings of the same phase space domain of $\mathbf V_{k}$ . The three different horizontal strips in the right hand partitioning refer to the three orthogonal subspaces $\mathbf O^+_{k+1}$ , $\mathbf V_{k+1}$ , and $\mathbf O^-_{k+1}$ .

As one can see from Figure 2.22, this corresponds to the union of the three strips mentioned above. Here the $P$ 's are the familiar o.n. wave packets defined by Eq.(2.61):

$$P_{1\ell}^{\varepsilon'/2}(t)= \frac{1}{\sqrt{2\pi}}\frac{1}{\sqrt{\varepsilon'/2}} \int^{\varep... ...}_{\varepsilon'/2} e^{-2\pi i\ell\omega/(\varepsilon'/2)} e^{i\omega t} d\omega$$ (289)

$$P_{-2\,\ell}^{\varepsilon'/2}(t)= \frac{1}{\sqrt{2\pi}}\frac{1}{\sqrt{\varepsilon'/2}} \int\limits^... ...{-\varepsilon'} e^{-2\pi i\ell\omega/(\varepsilon'/2)} e^{i\omega t} d\omega ~,$$ (290)

and

$$\Delta t = \frac{2\pi}{\varepsilon'/2}=2\times2^{(k+1)}$$

$$\Delta\omega = \varepsilon'/2=\frac{1}{2}\times\frac{2\pi}{2^{(k+1)}}$$

for both the positive and negative frequency wave packets. Due to the mutual orthogonality of all the $P$ 's and $Q$ 's combined, every band limited function $f\in\mathbf V_{k}$ is a unique linear combination of these elements. Thus we have identified two alternative bases of $\mathbf V_{k}$ . The first one consists of the elements exhibited by Eq.(2.87). The second one consists of the elements exhibited by Eqs.(2.88)-(2.90).

This fact is reexpressed by the statement that $\mathbf V_{k}$ is the direct sum of the subspaces

$$\mathbf O_{k+1}^+ \equiv span \{P_{1\ell}^{\varepsilon'/2}(t):\, \ell=0,\pm1,\cdots\}$$ (291)

$$\mathbf V_{k+1} \equiv span\{Q_{0\ell}^{\varepsilon'}(t):\, \ell=0,\pm1,\cdots\}$$ (292)

$$\mathbf O_{k+1}^- \equiv span\{P_{-2\,\ell}^{\varepsilon'/2}(t):\, \ell=0,\pm1,\cdots\}~,$$ (293)

or, symbolically, that

$$\mathbf V_{k}=\mathbf V_{k+1}\oplus\mathbf O_{k+1}^+\oplus\mathbf O_{k+1}^-~,$$

which is the same as Eq.(2.84), provided one sets

$$\mathbf O_{k+1}=\mathbf O_{k+1}^+\oplus\mathbf O_{k+1}^- ~,$$

the direct sum of the positive and negative frequency subspaces othogonal to the $(k+1)$ st resolution space $\mathbf V_{k+1}$ .