An Algebraic Theory of Wavelets. I. Operational Calculus and Complex Structure
Gerald Kaiser · SIAM Journal on Mathematical Analysis · 1992
In wavelet analysis, a function f is split into two parts at each iteration. The first part, $Hf$, represents a smoothed version of f, sampled half as frequently, while the second part, $Gf$, represents the detail lost by filtering through H. Although the operators H and G have very different interpretations, they exhibit a remarkable symmetry in their algebraic properties. We examine this symmetry by developing an effective, basis-independent operational calculus for wavelets and use it to show that the symmetry is due to the existence of a complex structure, i.e., a map J such that $J^2 = - I$ where I is the identity. This implies that the space $V_\alpha $ of (real) functions at the scale $2^\alpha (\alpha \in \mathbb{Z})$ may be regarded as a complexification$V_{\alpha + 1}^c $ of the space $V_{\alpha + 1} $ of functions at the next (coarser) scale. Roughly, the low-frequency parts $Hf$ of the functions span the real part of $V_{\alpha + 1}^c $ while their high-frequency parts $Gf$ span the imaginary part. The map J mediates between the two and relates the corresponding operators H and G. Furthermore, at the scale $\alpha = - 1$, J transforms the fundamental function $\phi $ associated with H into the “fundamental wavelet” $\psi $ associated with G.