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We will now look more closely at the basic scaling function and wavelet to see when they exist and what their properties are [link] , [link] , [link] , [link] , [link] , [link] , [link] . Using the same approach that is used in the theory of differentialequations, we will examine the properties of $\phi \left(t\right)$ by considering the equation of which it is a solution. The basic recursion [link] that comes from the multiresolution formulation is
with $h\left(n\right)$ being the scaling coefficients and $\phi \left(t\right)$ being the scaling function which satisfies this equation which is sometimes calledthe refinement equation , the dilation equation , or the multiresolution analysis equation (MRA).
In order to state the properties accurately, some care has to be taken in specifying just what classes of functions are being considered or areallowed. We will attempt to walk a fine line to present enough detail to be correct but not so much as to obscure the main ideas and results. A fewof these ideas were presented in Section: Signal Spaces and a few more will be given in the next section. A more complete discussion can be foundin [link] , in the introductions to [link] , [link] , [link] , or in any book on function analysis.
There are three classes of signals that we will be using. The most basic is called ${L}^{2}\left(\mathbf{R}\right)$ which contains all functions which have a finite, well-defined integral of the square: $f\in {L}^{2}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\Rightarrow \phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\int {\left|f\left(t\right)\right|}^{2}\phantom{\rule{0.166667em}{0ex}}dt=E<\infty $ . This class is important because it is a generalization of normal Euclideangeometry and because it gives a simple representation of the energy in a signal.
The next most basic class is ${L}^{1}\left(\mathbf{R}\right)$ , which requires a finite integral of the absolute value of the function: $f\in {L}^{1}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\Rightarrow \phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\int \left|f\left(t\right)\right|\phantom{\rule{0.166667em}{0ex}}dt=K<\infty $ . This class is important because one may interchange infinite summations and integrations with these functions although not necessarily with ${L}^{2}$ functions. These classes of function spaces can be generalized to those with ${\int \left|f\left(t\right)\right|}^{p}\phantom{\rule{0.166667em}{0ex}}dt=K<\infty $ and designated ${L}^{p}$ .
A more general class of signals than any ${L}^{p}$ space contains what are called distributions . These are generalized functions which are not defined by their having “values" but by the value of an “inner product"with a normal function. An example of a distribution would be the Dirac delta function $\delta \left(t\right)$ where it is defined by the property: $f\left(T\right)=\int f\left(t\right)\phantom{\rule{0.166667em}{0ex}}\delta (t-T)\phantom{\rule{0.166667em}{0ex}}dt$ .
Another detail to keep in mind is that the integrals used in these definitions are Lebesque integrals which are somewhat more general than the basic Riemann integral. The value of a Lebesque integral is notaffected by values of the function over any countable set of values of its argument (or, more generally, a set of measure zero). A function definedas one on the rationals and zero on the irrationals would have a zero Lebesque integral. As a result of this, properties derived using measuretheory and Lebesque integrals are sometime said to be true “almost everywhere," meaning they may not be true over a set of measure zero.
Many of these ideas of function spaces, distributions, Lebesque measure, etc. came out of the early study of Fourier series and transforms. It isinteresting that they are also important in the theory of wavelets. As with Fourier theory, one can often ignore the signal space classes and canuse distributions as if they were functions, but there are some cases where these ideas are crucial. For an introductory reading of this bookor of the literature, one can usually skip over the signal space designation or assume Riemann integrals. However, when a contradiction orparadox seems to arise, its resolution will probably require these details.
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