ABSTRACT
We apply the Lax-Phillips wave equation scattering theory to multiresolutions associated with wavelets. For wavelet scattering, the translation symmetry, the scaling operator, and the scaling function are identified in the scattering theoretic spectral transform; the scaling function is shown to be analytic; and an analytic spectral function is identified as an invariant for multiresolutions, normalized so that the Haar wavelet corresponds to the constant function. For the study of the functional equation, we introduce almost periodic spaces and establish a general convergence for the infinite product formula with the limit in the L 2-space of the corresponding Bohr group.
