NONSTATIONARY WAVELETS ON THE M-SPHERE FOR SCATTERED DATA

We construct classes of nonstationary wavelets generated by what we ca ll spherical basis functions, which comprise a subclass of Schoenberg' s positive definite functions on the m-sphere. The wavelets are intrin sically defined on the m-sphere and are independent of the choice of c oordinate system. In addition, they may be orthogonalized easily, if d esired. We will discuss decomposition, reconstruction, and localizatio n for these wavelets. In the special case of the 2-sphere, we derive a n uncertainty principle that expresses the trade-off between localizat ion and the presence of high harmonics-or high frequencies-in expansio ns in spherical harmonics. We discuss the application of this principl e to the wavelets that we construct. (C) 1996 Academic Press, Inc.