Simple universal bounds for Chebyshev-type quadratures Original Research Article

A Chebyshev-type quadrature for a probability measure σσ is a distribution which is uniform on nn points and has the same first kk moments as σσ. We give an upper bound for the minimal nn required to achieve a given degree kk, for σσ supported on an interval. In contrast to previous results of this type, our bound uses only simple properties of σσ 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 σσ. 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.