Simple Universal Bounds for Chebyshev-Type Quadratures
This work offers a general and simple method for bounding Chebyshev-type quadrature node counts, which is useful for researchers in numerical analysis and approximation theory.
The paper provides simple upper and lower bounds for the minimal number of nodes needed in Chebyshev-type quadratures for probability measures on an interval, using only basic properties and moment estimates. The bounds are shown to be sharp through examples, and applications to Gaussian quadrature, random node selection, and point sets on spheres and cylinders are discussed.
A Chebyshev-type quadrature for a probability measure sigma is a distribution which is uniform on n points and has the same first k moments as sigma. We give an upper bound for the minimal n required to achieve a given degree k, for sigma supported on an interval. In contrast to previous results of this type, our bound uses only simple properties of sigma 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 sigma. 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 can a Poisson process approximate certain continuous distributions. The paper concludes with a list of open questions.