Abstract
In this chapter, we introduce a far-reaching extension of spherical harmonics, in which the surface-area measure, the only rotation-invariant measure, on the sphere is replaced by a family of weighted measures invariant under a finite reflection group, and the Laplace operator is replaced by a sum of squares of Dunkl operators, a family of commuting first-order differential–difference operators. Our goal is to lay the foundation for developing weighted approximation and harmonic analysis on the sphere, which turn out to be indispensable for the corresponding theory, even for unweighted approximation and harmonic analysis, on the unit ball and on the simplex, as will be seen in later chapters.
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