A Functional Hilbert Space Approach to the Theory of Wavelets

We approach the theory of wavelets from the theory of functional Hilbert spaces. Starting with a Hilbert space H, we consider a subset V of H, for which the span is dense in H. We define a function of positive type on the index set I which labels the elements of V . This function of positive type induces uniquely a functional Hilbert space, which is a subspace of CI and there exists a unitary mapping from H onto this functional Hilbert space. Such functional Hilbert spaces, however, are not easily characterized. Next we consider a group G for the index set I and create the set V using a representation R of the group on H. The unitary mapping between H and the functional Hilbert space is easily recognized as the wavelet transform. We do not insist the representation to be irreducible and derive a generalization of the wavelet theorem as formulated by Grossmann, Morlet and Paul. The functional Hilbert space can in general not be identified with a closed subspace of L2(G), in contrast to the case of unitary, irreducible and square integrable representations. Secondly, we take for G a semi-direct product of two locally compact groups S o T , where S is abelian. In this case we give a more tangible description for the functional Hilbert space, which is easier to grasp. Finally, we provide an example where we take H = L2(R) and the Euclidean motion group for G. This example is inspired by an application of biomedical imaging, namely orientation bundle theory, which was the motivation for this report.