0.7 Generalizations of the basic multiresolution wavelet system  (Page 4/28)

and the limit after iteration is

assuming the product converges and $\text{Φ}\left(0\right)$ is well defined. This is a generalization of [link] and is derived in [link] .

Properties of m-band wavelet systems

These theorems, relationships, and properties are generalizations of those given in Chapter: The Scaling Function and Scaling Coefficients, Wavelet and Wavelet Coefficients and Section: Further Properties of the Scaling Function and Wavelet with some outline proofs or derivations given in the Appendix. For the multiplicity- $M$ problem, if the support of the scaling function and wavelets and their respective coefficientsis finite and the system is orthogonal or a tight frame, the length of the scaling function vector orfilter $h\left(n\right)$ is a multiple of the multiplier $M$ . This is $N=MG$ , where Resnikoff and Wells [link] call $M$ the rank of the system and $G$ the genus.

The results of [link] , [link] , [link] , and [link] become

Theorem 28 If $\phi \left(t\right)$ is an $L^1$ solution to [link] and $\int \phi \left(t\right)dt\ne 0$ , then

This is a generalization of the basic multiplicity-2 result in [link] and does not depend on any particular normalization or orthogonality of $\phi \left(t\right)$ .

Theorem 29 If integer translates of the solution to [link] are orthogonal, then

This is a generalization of [link] and also does not depend on any normalization. An interesting corollary of this theorem is

Corollary 3 If integer translates of the solution to [link] are orthogonal, then

A second corollary to this theorem is

Corollary 4 If integer translates of the solution to [link] are orthogonal, then

This is also true under weaker conditions than orthogonality as was discussed for the $M=2$ case.

Using the Fourier transform, the following relations can be derived:

Theorem 30 If $\phi \left(t\right)$ is an $L^1$ solution to [link] and $\int \phi \left(t\right)dt\ne 0$ , then

which is a frequency domain existence condition.

Theorem 31 The integer translates of the solution to [link] are orthogonal if and only if

Theorem 32 If $\phi \left(t\right)$ is an $L^1$ solution to [link] and $\int \phi \left(t\right)dt\ne 0$ , then

if and only if

This is a frequency domain orthogonality condition on $h\left(n\right)$ .

Corollary 5

which is a generalization of [link] stating where the zeros of $H\left(\text{ω}\right)$ , the frequency response of the scaling filter, are located. This is an interestingconstraint on just where certain zeros of $H\left(z\right)$ must be located.

Theorem 33 If ${\sum }_{n}h\left(n\right)=\sqrt{M}$ , and $h\left(n\right)$ has finite support or decays fast enough, then a $\phi \left(t\right)\in L^2$ that satisfies [link] exists and is unique.

Theorem 34 If ${\sum }_{n}h\left(n\right)=\sqrt{M}$ and if ${\sum }_{n}h\left(n\right)h(n-Mk)=\delta \left(k\right)$ , then $\phi \left(t\right)$ exists, is integrable, and generates a wavelet system that is a tight frame in $L^2$ .

These results are a significant generalization of the basic $M=2$ wavelet system that we discussed in the earlier chapters. The definitions, properties, andgeneration of these more general scaling functions have the same form as for $M=2$ , but there is no longer a single wavelet associated with the scaling function. There are $M-1$ wavelets. In addition to [link] we now have $M-1$ wavelet equations, which we denote as

for

Some authors use a notation $h_0\left(n\right)$ for $h\left(n\right)$ and ${\phi }_0\left(t\right)$ for $\psi \left(t\right)$ , so that $h_{\ell }\left(n\right)$ represents the coefficients for the scaling function and all the wavelets and ${\phi }_{\ell }\left(t\right)$ represents the scaling function and all the wavelets.