TABLE OF CONTENTS The purpose of this appendix is to present several of the basic theorems about wavelet transforms that have been used in the text but that have not been proved. The techniques used to establish these results are typical of those used in wavelet theory, and the proofs will illustrate where the different assumptions about the analyzing wavelet are used.
To simplify the notation, we present the results in one dimension, although the results are true for 
. We assume throughout this section that the analyzing wavelet ? is in 
and that the wavelets are defined by
For future reference, we note that the mapping ( a,b) ? ? (a,b) is continuous from 
.
The wavelet transform is defined for 
by
where, as elsewhere, ? (a,b) (x) = ? (a,b) (x). Then W f(a,b) ? ? f ? ???, and thus by our remark about the continuity of (a,b) ? ? (a,b), it is clear that W f(a,b) is continuous on 
.
We wish to prove that the mapping f ? W f(a,b) is a partial isometry from 
into 
. This is not true in general, so additional assumptions must be made about ?. The assumption we make, which is called an admissibility condition,is that
for almost all 
. We have written the admissibility condition so that it is clear how the results generalize to 
.
