Wavelet basis and Wavelet subspace

Considering the optimized family of wavelets ${\bf\Psi }$, its closed linear span constitutes a subspace $<{\bf\Psi}>$ in the ${\Bbb{L}}^2({\Bbb{R}}^2)$ space. With eq. (8) any function can be orthogonally projected into that subspace. It is interesting to ask whether ${\bf\Psi }$ constitutes a basis, because then the projection is unique. That this is indeed so can be shown with induction over the number of wavelets: Consider $n$ wavelets $(\psi_{{\bf n}_1},\ldots,\psi_{{\bf n}_n})$, that minimize the energy functional (2) and that form a basis. Let us choose a new wavelet that approximates best the residual between the function $f$ and its approximation with the first $n$ wavelets. After optimization of the $n$th$+ 1$ wavelet, the energy functional (2) is smaller than before (for the $n$ wavelets):

$$\Vert f-\sum_{i=1}^n w_i \psi_{{\bf n}_i}\Vert > \Vert f-\sum_{i=1}^n w_i \psi_{{\bf n}_i} - w_{n+1} \psi_{{\bf n}_{n+1}}\Vert\; .$$

Assuming now, that

$$<\psi_{{\bf n}_1},\ldots,\psi_{{\bf n}_n}> = <\psi_{{\bf n}_1},\ldots,\psi_{{\bf n}_{n+1}}>$$

we have in particular

$$<\psi_{{\bf n}_1},\ldots,\psi_{{\bf n}_n}>^{\perp} = <\psi_{{\bf n}_1},\ldots,\psi_{{\bf n}_{n+1}}>^{\perp}.$$

This again means that

$$f-\sum_{i=1}^{n} w_i \psi_{{\bf n}_i} \in (<\psi_{{\bf n}_1},\ldots,\psi_{{\bf n}_{n+1}}>)^{\perp}\; ,$$

which implies

$$\langle f-\sum_{i=1}^{n} w_i \psi_{{\bf n}_i},\psi_{{\bf n}_{n+1}} \rangle = 0 \; .$$

This, however, contradicts the choice of $\psi_{{\bf n}_n}$ in the optimization step, where $\psi_{{\bf n}_n}$ was selected such that

$$\langle f-\sum_{i=1}^{n} w_i \psi_{{\bf n}_i},\psi_{{\bf n}_{n+1}} \rangle \neq 0 \; .$$

Let us call the closed linear span of $<{\bf\Psi}>$ ${\Bbb{L}}^2({\Bbb{R}}^2)$ (image) subspace. The dual wavelets $\tilde{\psi}_{{\bf n}_i}$ are linearly independent, and the projection

$${\bf w}= \tilde{{\bf\Psi}}g$$

establishes an isomorphism from ${\Bbb{L}}^2({\Bbb{R}}^2)$ (or the image space, respectively) into $\Bbb{R}^n$ which is the space of the $n$-vectors containing the wavelet coefficients. This space is dual to the image subspace and we call it the wavelet subspace.

Figure: A function $g \in {\Bbb{L}}^2({\Bbb{R}}^2)$ is mapped by the linear mapping $\tilde{{\bf \Psi}}$ onto the vector ${\bf w}\in \Bbb{R}^N$ in the wavelet subspace. The mapping of ${\bf w}$ into ${\Bbb{L}}^2({\Bbb{R}}^2)$ is achieved with the linear mapping ${\bf \Psi}$. Both mappings constitute an orthogonal projection of a function $g \in {\Bbb{L}}^2({\Bbb{R}}^2)$ into the (image) subspace $<{\bf \Psi}> \subset {\Bbb{L}}^2({\Bbb{R}}^2)$.