Simple universal bounds for Chebyshev-type quadratures

A Chebyshev-type quadrature for a probability measure σ is a distribution which is uniform on n points and has the same first k moments as σ. We give bounds for the smallest possible n required to achieve a certain degree k. In contrast to previous results of this type, our bounds use only simple properties of σ and are thus applicable in wide generality. In particular, it is shown that whenever σ has bounded density on a finite interval, n may increase at most exponentially with k. Examples are given illustrating the tightness of our bounds, and applications are given to special local constructions on the sphere and cylinder and to an apparently new result on Gaussian quadrature. We also introduce the concept of random Chebyshev-type quadratures, the case in which nodes are chosen by independent random samples from σ. The concept is discussed and some preliminary results are proven. These results were recently applied to understand how well can a Poisson process approximate certain continuous distributions. We conclude with a list of open questions.