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Approximation by Means of h-Harmonic Polynomials on the Unit Sphere

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Abstract

The h-harmonics are analogues of the ordinary harmonics, they are orthogonal homogeneous polynomials on the sphere with respect to a weight function that is invariant under a reflection group. Two means of associated orthogonal expansions, the de la Vallée Poussin means and an analog of spherical means, are defined and their approximation behaviors are studied. A weighted modulus of smoothness is defined using the modified spherical means and is proved to be equivalent to a weighted K-modulus defined using the differential-difference h-spherical Laplacian. A Bernstein type inequality for the h-spherical Laplacian is also established.

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Authors and Affiliations

  1. Department of Mathematics, University of Oregon, Eugene, OR, 97403-1222, USA

    Yuan Xu

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  1. Yuan Xu
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Xu, Y. Approximation by Means of h-Harmonic Polynomials on the Unit Sphere. Advances in Computational Mathematics 21, 37–58 (2004). https://doi.org/10.1023/B:ACOM.0000016433.27278.2a

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  • DOI: https://doi.org/10.1023/B:ACOM.0000016433.27278.2a

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