Asymptotic positivity of multivariate product-integration weights

Determine whether the quadrature weights produced by the multivariate kernel-based method with the periodizing transformation and Chebyshev density possess the asymptotic positivity property, namely whether the sum of their absolute values converges to the sum of the weights as the number of points tends to infinity.

Background

The paper interprets its quadrature construction as a multivariate analogue of product integration: translated kernel functions approximate the smooth factor, while integration against the nonsmooth density determines the quadrature weights. In classical product-integration settings, such as interpolation at Clenshaw–Curtis points or Jacobi-polynomial zeros with a nonnegative weight function, the weights may exhibit asymptotic positivity.

Asymptotic positivity is defined by the limiting relation between the sum of the weights and the sum of their absolute values. If it holds, the total contribution of negative weights becomes negligible as the number of points increases, providing a desirable stability property. The paper explicitly states that whether this property holds for the present multivariate method is unresolved.