The first summary is given in four tables of the basic
relationships and equations, primarily developed in Chapter: The Scaling Function and Scaling Coefficients, Wavelet and Wavelet Coefficients , for
the scaling function
Case  Condition 


Signal Space 
1  Multiresolution 


distribution 
2  Partition of 1 


distribution 
3  Orthogonal 



5  SF Smoothness 

poly 

6  SF Moments 

Coiflets 
Case  Condition 


Eigenval.{T} 
1  Existence 



2  Fundamental 



3  QMF 



4  Orthogonal 


one 
and 
others 

6  Coiflets 

Case  Condition 


Signal Space 
1  MRA 


distribution 
3  Orthogonal 



3  Orthogonal 



5  W Moments 

poly 
Case  Condition 


Eigenval.{T} 
2  Fundamental 



3  Orthogonal 



3  Orthogonal 



5  Smoothness 



The different “cases" represent somewhat similar conditions for
the stated relationships. For example, in Case 1, Table 1, the
multiresolution conditions are stated in the time and frequency
domains while in Table 2 the corresponding necessary conditions
on
If we “tighten" the restrictions by adding one more linear
condition, we get Case 2 which has consequences in Tables 1, 2,
and 4 but does not guarantee anything better that a
distribution. Case 3 involves orthogonality, both across scales
and translations, so there are two rows for Case 3 in the tables
involving wavelets. Case 4 adds to the orthogonality a
condition on the frequency response
In some cases, columns 3 and 4 are equivalent and others, they are not. In some categories, a higher numbered case assumes a lower numbered case and in others, they do not. These tables try to give a structure without the details. It is useful to refer to them while reading the earlier chapters and to refer to the earlier chapters to see the assumptions and conditions behind these tables.