Titlebook: Approximation Theory and Harmonic Analysis on Spheres and Balls; Feng Dai,Yuan Xu Book 2013 Springer Science+Business Media New York 2013

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Approximation on the Sphere,, in terms of the smoothness of the function. In this chapter, we study the characterization of the best approximation by polynomials on the sphere. In the classical setting of one variable, the smoothness of a function on . is described by the modulus of smoothness, defined via the forward differen

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Weighted Polynomial Inequalities,be established in this chapter. Since some of them will be needed in weighted approximation theory and harmonic analysis in later chapters, we prove them in the weighted .. norm. We will work in the context of doubling weights, defined and discussed in the first section. A fundamental tool in our ap

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Cubature Formulas on Spheres,ocesses of approximation. Cubature formulas, a synonym for numerical integration formulas, are essential tools for discretizing integrals. In contrast to the one-variable case, fundamental problems of cubature formulas in several variables are still open, including those on the sphere. In this chapt

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Harmonic Analysis Associated with Reflection Groups, 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 we

squander

,Boundedness of Projection Operators and Cesàro Means,underlying structure. In this chapter, we establish the boundedness of the Cesáro means for .-harmonic expansions with respect to the product weights ...(.) = ... | .. | .. on the sphere. The main results are stated and discussed in the first section. The central piece of the proof is a pointwise es

歹徒

,Projection Operators and Cesàro Means in ,, Spaces, the critical index ., and furthermore, they are bounded under the same condition as that of the Bochner–Riesz means. In this chapter, we establish such results for .-harmonic expansions with respect to the product ., which cover results for ordinary spherical harmonic expansions. The proof of such

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