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Nonstationary wavelets on the \(m\)-sphere for scattered data. (English) Zbl 0858.42025
Summary: We construct classes of nonstationary wavelets generated by what we call spherical basis functions, which comprise a subclass of Schoenberg’s positive definite functions on the \(m\)-sphere. The wavelets are intrinsically defined on the \(m\)-sphere and are independent of the choice of coordinate system. In addition, they may be orthogonalized easily, if desired. We discuss decomposition, reconstruction, and localization for these wavelets. In the special case of the 2-sphere, we derive an uncertainty principle that expresses the tradeoff between localization and the presence of high harmonics – or high frequencies – in expansions in spherical harmonics. We discuss the application of this principle to the wavelets that we construct.

42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
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