# Well Conditioned Spherical Designs for Integration and Interpolation on the Two-Sphere

@article{An2010WellCS, title={Well Conditioned Spherical Designs for Integration and Interpolation on the Two-Sphere}, author={Congpei An and Xiaojun Chen and Ian H. Sloan and Robert S. Womersley}, journal={SIAM J. Numer. Anal.}, year={2010}, volume={48}, pages={2135-2157} }

A set $\mathcal{X}_{N}$ of $N$ points on the unit sphere is a spherical $t$-design if the average value of any polynomial of degree at most $t$ over $\mathcal{X}_{N}$ is equal to the average value of the polynomial over the sphere. This paper considers the characterization and computation of spherical $t$-designs on the unit sphere $\mathbb{S}^2\subset\mathbb{R}^3$ when $N\geq(t+1)^2$, the dimension of the space $\mathbb{P}_t$ of spherical polynomials of degree at most $t$. We show how to…

## 44 Citations

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The worst-case errors of quadrature rules using spherical $t_\epsilon$-designs in a Sobolev space are studied, and a model of polynomial approximation with the $l_1$-regularization using spherical £t_Â£t- designs is investigated.

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A spherical $t$-design is a set of points on the sphere that are nodes of a positive equal weight quadrature rule having algebraic accuracy $t$ for all spherical polynomials with degrees $\le t$.…

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In this paper we make a comparison between certain probabilistic and deterministic point sets and show that some deterministic constructions (spherical $t$-designs) are better or as good as…

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This talk presents some joint work with An, Frommer, Lang, Sloan and Womersley on spherical designs and polynomial approximation on the sphere [1],[2],[4],[5]. Finding “good” finite sets of points on…

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Cohn--Conway--Elkies--Kumar [Experiment. Math. (2007)] described that one can construct a family of designs on $S^{2n-1}$ from a design on $\mathbb{CP}^{n-1}$. In this paper, we prove their claim for…

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