Multivariate Christoffel functions and hyperinterpolation

We obtain upper bounds for Lebesgue constants (uniform norms) of hyperinterpolation operators via estimates for (the reciprocal of) Christoffel functions, with different measures on the disk and ball, and on the square and cube. As an application, we show that the Lebesgue constant of total-degree polynomial interpolation at the Morrow-Patterson minimal cubature points in the square has an $\mathcal{O}(\mbox{deg}^3)$ upper bound, explicitly given by the square root of a sextic polynomial in the degree.