0.5 The scaling function and scaling coefficients, wavelet and  (Page 9/13)

The basic recursive equation for the scaling function, defined in [link] as

is homogeneous, so its solution is unique only within a normalization factor. In most cases, both the scaling function and wavelet arenormalized to unit energy or unit norm. In the properties discussed here, we normalize the energy as $E={\int \left|\phi \left(t\right)\right|}^{2}dt=1.$ Other normalizations can easily be used if desired.

General properties not requiring orthogonality

There are several properties that are simply a result of the multiresolution equation [link] and, therefore, hold for orthogonal and biorthogonal systems.

Property 1 The normalization of $\phi \left(t\right)$ is arbitrary and is given in [link] as $E$ . Here we usually set $E=1$ so that the basis functions are orthonormal and coefficients can easily be calculated with inner products.

Property 2 Not only can the scaling function be written as a weighted sum of functions in the next higher scale space as stated in the basic recursion equation [link] , but it can also be expressed in higher resolution spaces:

where $h^{\left(1\right)}\left(n\right)=h\left(n\right)$ and for $j\ge 1$

Property 3 A formula for the sum of dyadic samples of $\phi \left(t\right)$

Property 4 A “partition of unity" follows from [link] for $J=0$

Property 5 A generalized partition of unity exists if $\phi \left(t\right)$ is continuous

Property 6 A frequency domain statement of the basic recursion equation [link]

Property 7 Successive approximations in the frequency domain is often easier to analyze than the time domain version in [link] . The convergence properties of this infinite product are very important.

This formula is derived in [link] .

Properties that depend on orthogonality

The following properties depend on the orthogonality of the scaling and wavelet functions.

Property 8 The square of the integral of $\phi \left(t\right)$ is equal to the integral of the square of $\phi \left(t\right)$ , or $A_0^2=E$ .

Property 9 The integral of the wavelet is necessarily zero

The norm of the wavelet is usually normalized to one such that ${\int \left|\psi \left(t\right)\right|}^{2}dt=1$ .

Property 10 Not only are integer translates of the wavelet orthogonal; different scales are also orthogonal.

where the norm of $\psi \left(t\right)$ is one.

Property 11 The scaling function and wavelet are orthogonal over both scale and translation.

for all integer $i,j,k,\ell$ where $j\le i$ .

Property 12 A frequency domain statement of the orthogonality requirements in [link] . It also is a statement of equivalent energy measures in the time and frequency domains as in Parseval's theorem, which is true with anorthogonal basis set.

Property 13 The scaling coefficients can be calculated from the orthogonal or tight frame scaling functions by

Property 14 The wavelet coefficients can be calculated from the orthogonal or tight frame scaling functions by

Derivations of some of these properties can be found in Appendix B . Properties in equations [link] , [link] , [link] , [link] , [link] , [link] , and [link] are independent of any normalization of $\phi \left(t\right)$ . Normalization affects the others. Those in equations [link] , [link] , [link] , [link] , [link] , [link] , and [link] do not require orthogonality of integer translates of $\phi \left(t\right)$ . Those in [link] , [link] , [link] , [link] , [link] , [link] , [link] require orthogonality. No properties require compact support. Many of the derivations interchangeorder of summations or of summation and integration. Conditions for those interchanges must be met.