Continuous wavelet transforms with applications to analyzing functions on spheres
The paper provides a generalized continuous wavelet transform for \(L^2\)-spaces on a compact manifold \(M\). The generalized translation can be defined as the transitive action of a Lie group \(G\) on \(M\). However, the scaling process does not fit together with the topological structure of the manifold. Therefore one works with the tangent bundle \(TM\) of \(M\). Then the scaling process can be performed in the (noncompact) fibers. The idea is worked out for \(TS^1\) and \(TS^2\). The paper follows the general construction of wavelet transforms by \textit{A. Grossmann, J. Morlet} and \textit{T. Paul} [J. Math. Phys. 26, 2473-2479 (1985; Zbl 0571.22021)], looking for groups with square-integrable representations in \(L^2(TS^1)\) or \(L^2(TS^2)\). Applications of this construction for analyzing functions on \(S^1\) and \(S^2\) are presented.
scientific article; zbMATH DE number 1153111