Simple Universal Bounds for Chebyshev-Type Quadratures

TL;DR

This paper establishes simple, general bounds on the number of nodes needed for Chebyshev-type quadratures for measures supported on intervals, with applications to Gaussian quadrature and point set constructions.

Contribution

It provides new upper and lower bounds for the minimal number of nodes in Chebyshev-type quadratures using simple properties of the measure, applicable in broad settings.

Findings

Derived an upper bound for the number of nodes based on measure properties.

Established a lower bound using moment estimates.

Applied bounds to construct point sets on spheres and cylinders.

Abstract

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.