On weighted Chebyshev type quadrature formulas
Mathematics of Computation
Bounds for the number of nodes in Chebyshev type quadrature formulas
Journal of Approximation Theory
On a new method for constructing good point sets on spheres
Discrete & Computational Geometry
Chebyshev-type quadrature on multidimensional domains
Journal of Approximation Theory
Chebyshev-type quadrature for Jacobi weight functions
Proceedings of the fourth international symposium on Orthogonal polynomials and their applications
Numerical Cubature from Archimedes' Hat-box Theorem
SIAM Journal on Numerical Analysis
New asymptotic estimates for spherical designs
Journal of Approximation Theory
Spherical Designs via Brouwer Fixed Point Theorem
SIAM Journal on Discrete Mathematics
New computationally efficient quadrature formulas for pyramidal elements
Finite Elements in Analysis and Design
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A Chebyshev-type quadrature for a probability measure @s is a distribution which is uniform on n points and has the same first k moments as @s. We give an upper bound for the minimal n required to achieve a given degree k, for @s supported on an interval. In contrast to previous results of this type, our bound uses only simple properties of @s and is applicable in wide generality. We also obtain a lower bound for the required number of nodes which only uses estimates on the moments of @s. Examples illustrating the sharpness of our bounds are given. As a corollary of our results, we obtain an apparently new result on the Gaussian quadrature. In addition, we suggest another approach to bounding the minimal number of nodes required in a Chebyshev-type quadrature, utilizing a random choice of the nodes, and propose the challenge of analyzing its performance. A preliminary result in this direction is proved for the uniform measure on the cube. Finally, we apply our bounds to the construction of point sets on the sphere and cylinder which form local approximate Chebyshev-type quadratures. These results were needed recently in the context of understanding how well a Poisson process can approximate certain continuous distributions. The paper concludes with a list of open questions.