Are there any $1$-dimensional quadrature rules of arbitrary accuracy, on either $[0,1]$ or $\mathbb{R}$, with any non-trivial weight function, such that the associated $N$-dimensional cubature rule constructed using the Smolyak (sparse grid) construction has positive weights for all $N\in\mathbb{N}$?
If not, are there variants of the Smolyak construction that guarantee this?
I know that I can always produce a positive weighted rule using the tensor product construction. The question is whether there are positive weighted rules with significantly fewer nodes than under the tensor product construction.
This question is related to, but more general than, this one: https://math.stackexchange.com/questions/2068696/generating-sparse-grids-with-non-negative-weights
This thesis ("Smolyak Quadrature", by Vesa Kaarnioja, University of Helsinki) provides an excellent background to the Smolyak construction for those interested: https://helda.helsinki.fi/bitstream/handle/10138/40159/thesis.pdf
