Appendix B: Wavelet Transforms
2001
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. B.1 The L2 theory To simplify the notation, we present the results in one dimension, although the results are true for ℝn. We assume throughout this section that the analyzing wavelet ψ is in L2(ℝ) and that the wavelets are defined by ψ (a,b) (x) = 1 a ψ ( x−b a ) , a>0,b ∈ ℝ. B.1 For future reference, we note that the mapping (a, b) ↦ ψ(a,b) is continuous from ℝ* × R to L2(ℝ).