Resolution Spaces as Hierarchical

Third, take note of the hierarchical subspace structure of the resolution spaces $\mathbf V_{k},\,k=\cdots,-1,0,1,\cdots$ . The Fourier transform of the basis elements, Eq.(2.90), for $\mathbf V_{k}$ have compact support confined to $[-\pi2^{-k},\pi2^{-k}]$ . As was shown in Part (c) of Ex. 1.5.3 on page , these basis elements form a complete set. This means that $f\in\mathbf V_{k}$ if and only if its Fourier transform has support confined to $[-\pi2^{-k},\pi2^{-k}]$ . Next consider the vector space $\mathbf V_{k+1}$ . The Fourier transform of its basis elements have support confined to

$$\left[-\pi2^{-(k+1)},\pi2^{-(k+1)}\right]\subset\left[-\pi2^{-k},\pi2^{-k}\right]$$

In fact, every element of $\mathbf V_{k+1}$ enjoys this property. This implies that such elements also belong to $\mathbf V_{k}$ . Thus one has

$$f\in\mathbf V_{k+1}\Rightarrow f\in\mathbf V_{k}~.$$

In other words, $\mathbf V_{k+1}$ is a subspace of $\mathbf V_{k}$ :

$$\mathbf V_{k+1}\subset \mathbf V_{k}~.$$

More explicitly, this inclusion property says that

$$\{0\}\subset\cdots\subset\mathbf V_{2}\subset\mathbf V_{1}\subset... ...\mathbf V_{-1} \subset\mathbf V_{-2}\subset\cdots\subset L^2(-\infty,\infty)~.$$

Such a hierarchy of increasing subspaces is called a multiscale analysis of the space of square-integrable functions. A multiscale analysis is always derived from (i.e. based on) a scaling function $\phi$ . In our illustrative example this scaling function (``father function'') is Shannon's sinc function, Eq.(2.88).